∗

A Quantum Engineer’s Guide to Superconducting QubitsP. Krantz1,2,†, M. Kjaergaard1, F. Yan1, T.P. Orlando1, S. Gustavsson1, and W. D. Oliver1,3,‡1Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139,USA2Wallenberg Centre for Quantum Technology (WACQT), Chalmers University of Technology, Gothenburg, SE-41296,Sweden and3MIT Lincoln Laboratory, 244 Wood Street, Lexington, MA 02420, USA

(Dated: 12 August 2019)

The aim of this review is to provide quantum engineers with an introductory guide to the central conceptsand challenges in the rapidly accelerating field of superconducting quantum circuits. Over the past twentyyears, the field has matured from a predominantly basic research endeavor to one that increasingly exploresthe engineering of larger-scale superconducting quantum systems. Here, we review several foundationalelements – qubit design, noise properties, qubit control, and readout techniques – developed during thisperiod, bridging fundamental concepts in circuit quantum electrodynamics (cQED) and contemporary, state-of-the-art applications in gate-model quantum computation.

CONTENTS

I. Introduction 2A. Organization of this article 2

II. Engineering quantum circuits 3A. From quantum harmonic oscillator to the

transmon qubit 3B. Qubit Hamiltonian engineering 6

1. Tunable qubit: split transmon 62. Towards larger anharmonicity: flux qubit

and fluxonium 7C. Interaction Hamiltonian engineering 9

1. Physical coupling: capacitive andinductive 9

2. Coupling axis: transverse andlongitudinal 10

III. Noise, decoherence, and error mitigation 11A. Types of noise 11

1. Systematic noise 122. Stochastic noise 123. Noise strength and qubit susceptibility 12

B. Modeling noise and decoherence 121. Bloch sphere representation 122. Bloch-Redfield model of decoherence 133. Modification due to 1/f -type noise 164. Noise power spectral density (PSD) 17

C. Common examples of noise 171. Charge noise 172. Magnetic flux noise 183. Photon number fluctuations 184. Quasiparticles 18

D. Operator form of qubit-environmentinteraction 19

∗†philipk@mit.edu,‡william.oliver@mit.edu

1. Connecting T1 to S(ω) 192. Connecting Tϕ to S(ω) 193. Noise spectroscopy 21

E. Engineering noise mitigation 211. Materials and fabrication improvements 212. Design improvements 213. Dynamical error suppression 224. Cryogenic engineering 22

IV. Qubit control 22A. Boolean logic gates used in classical

computers 22B. Quantum logic gates used in quantum

computers 23C. Comparing classical and quantum gates 26

1. Gate sets and gate synthesis 262. Addressing superconducting qubits 27

D. Single-qubit gates 271. Capacitive coupling for X,Y control 272. Virtual Z gate 293. The DRAG scheme 30

E. The iSWAP two-qubit gate in tunablequbits 311. Deriving the iSWAP unitary 312. Applications of the iSWAP gate 32

F. The CPHASE two-qubit gate in tunablequbits 321. Trajectory design for the CPHASE gate 342. The CPHASE gate for quantum error

correction 343. Quantum simulation and algorithm

demonstrations using CPHASE 35G. Two-qubit gates using only microwaves 36

1. The operational principle of the CR gate 362. Improvements to the CR gate and

quantum error correction experimentsusing CR 37

3. Quantum simulation and algorithmdemonstrations with the CR gate 37

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4. Other microwave-only gates: bSWAP,MAP, and RIP 38

H. Gate implementations with tunablecoupling 39

V. Qubit readout 40A. Dispersive readout 40B. Measuring the resonator amplitude and

phase 421. Representation of the readout signal 432. I-Q mixing 433. Homodyne demodulation 434. Heterodyne demodulation 45

C. Weak and strong qubit measurements:Impact of noise 46

D. “Purcell filters” for faster readout 47E. Improve signal-to-noise ratio: Parametric

amplification 481. Quantum-limited amplification processes 492. Operation of Josephson parametric

amplifiers 503. The traveling wave parametric amplifier 52

VI. Summary and outlook 53

Acknowledgments 54

I. INTRODUCTION

Quantum processors harness the intrinsic properties ofquantum mechanical systems – such as quantum paral-lelism and quantum interference – to solve certain prob-lems where classical computers fall short1–6. Over thepast two decades, rapid developments in the science andengineering of quantum systems have advanced the fron-tier in quantum computation, from the realm of scien-tific explorations on single isolated quantum systems to-ward the creation and manipulation of multi-qubit pro-cessors7,8. In particular, the requirements imposed bylarger quantum processors have shifted of mindset withinthe community, from solely scientific discovery to the de-velopment of new, foundational engineering abstractionsassociated with the design, control, and readout of multi-qubit quantum systems. The result is the emergence of anew discipline termed quantum engineering, which servesto bridge the basic sciences, mathematics, and computerscience with fields generally associated with traditionalengineering.

One prominent platform for constructing a multi-qubitquantum processor involves superconducting qubits, inwhich information is stored in the quantum degrees offreedom of nanofabricated, anharmonic oscillators con-structed from superconducting circuit elements. In con-trast to other platforms, e.g. electron spins in sili-con9–14 and quantum dots15–18, trapped ions19–23, ul-tracold atoms24–27, nitrogen-vacancies in diamonds28,29,

and polarized photons30–33, where the quantum informa-tion is encoded in natural microscopic quantum systems,superconducting qubits are macroscopic in size and litho-graphically defined.

One remarkable feature of superconducting qubits isthat their energy-level spectra are governed by circuit el-ement parameters and thus are configurable; they can bedesigned to exhibit “atom-like” energy spectra with de-sired properties. Therefore, superconducting qubits arealso often referred to as artificial atoms, offering a richparameter space of possible qubit properties and opera-tion regimes, with predictable performance in terms oftransition frequencies, anharmonicity, and complexity.

While there are many other excellent reviews on su-perconducting qubits, see e.g. Refs. 34–43, this workspecifically aims to introduce new quantum engineers(academic and industrial alike) to the terminology andstate-of-the-art practices used in the rapidly accelerat-ing field of superconducting quantum computing. Thereader is assumed to be familiar with basic concepts thatspan classical physics, quantum mechanics, and electricalengineering. In particular, readers will find it useful tohave had previous exposure to classical mechanics, theSchrodinger equation, the Bloch sphere representation ofqubit states, second quantization, basic concepts of su-perconductivity, electromagnetism, introductory circuitanalysis, classical boolean logic, linear dynamical sys-tems, analog and digital signal processing, and famil-iarity with microwave components such as transmissionlines and mixers. These topics will be introduced as theyarise, but having basic prior knowledge will be helpful.

A. Organization of this article

This review is organized in the following four sections;first, in Sec. II, we explore the parameter space avail-able when designing superconducting circuits. In particu-lar, we look at the promising capacitively-shunted planarqubit modalities and how these can be engineered withdesired properties, such as transition frequency, anhar-monicity, and reduced susceptibility to various sourcesof noise. In this section, we also introduce several waysin which interactions between qubits can be engineered,in order to implement two-qubit entangling operations,needed for a universal gate set.

In Sec. III, we discuss systematic and stochastic noise,the concepts of noise strength and qubit noise suscepti-bility, and the common sources of noise which lead todecoherence in superconducting circuits. We introducethe Bloch-Redfield model of decoherence, characterizedby longitudinal and transverse relaxation times T1 andT2, and discuss the implications of 1/f noise. We thendefine the noise power spectral density, which is com-monly used to characterize noise processes, and describehow it drives decoherence. Finally, we close the sectionwith a review of coherent control methods used to miti-gate certain types of coherence, reversible noise.

3

In Sec. IV, we provide a review of how single- andtwo-qubit operations are typically implemented in super-conducing circuits, by using a combination of local mag-netic flux control and microwave drives. In particular, wediscuss the family of two-qubit gates arising from a ca-pacitive coupling between qubits, and introduce severalrecent advances that have been demonstrated to achievehigh-fidelity gates, as well as applications in quantuminformation processing that use these gates. The contin-ued development of high-fidelity two-qubit gates in super-conducting qubits is a highly active research area. Forthis reason, we include sufficient technical details that areader may use this review as a starting point to criticallyassess the pros and cons of the various gates, as well asdevelop an appreciation for the types of gate-engineeringalready implemented in state-of-the-art superconductingquantum processors.

Finally, in Sec. V, we discuss the physics and engi-neering associated with the dispersive readout technique,typically used to measure the individual qubit states inmodern quantum processors. After a discussion of thetheory behind dispersive coupling, we give an introduc-tion to design of Purcell filters and the development ofquantum-limited parametric amplifiers.

II. ENGINEERING QUANTUM CIRCUITS

In this section, we will demonstrate how quantum sys-tems based on superconducting circuits can be engineeredto achieve certain desired properties. Using the mostcommon qubit modalities, we discuss how properties suchas the qubit transition frequency, anharmonicity, andnoise susceptibility can be tailored by the choice of circuittopology and element parameter values. We also discusshow to engineer the interactions between different quan-tum systems, in particular the cases of qubit-qubit andqubit-resonator couplings.

A. From quantum harmonic oscillator to the transmonqubit

A quantum mechanical system is governed by the time-dependent Schrodinger equation,

H|ψ(t)〉 = i~∂

∂t|ψ(t)〉, (1)

where |ψ(t)〉 is the state of the quantum system at timet, ~ is the reduced Planck’s constant h/2π, and H is theHamiltonian that describes the total energy of the sys-tem. The “hat” is used to indicate that H is a quantumoperator. As the Schrodinger equation is a first-order lin-ear differential equation, the temporal dynamics of thequantum system may be viewed as a straightforward ex-ample of a linear dynamical system with formal solution,

|ψ(t)〉 = e−iHt/~|ψ(0)〉. (2)

The time-independent Hamiltonian H governs the timeevolution of the system through the operator e−iHt/~.Thus, just as with classical systems, determining theHamiltonian of a system – whether the classical Hamil-tonian H or its quantum counterpart H – is the first stepto deriving its dynamical behavior. In Sec. IV, we con-sider the case when the Hamiltonian is time-dependentin the context of qubit control.

To understand the dynamics of a superconductingqubit circuit, it is natural to start with the classical de-scription of a linear LC resonant circuit [Fig. 1(a)]. Inthis system, energy oscillates between electrical energyin the capacitor C and magnetic energy in the inductorL. In the following, we will arbitrarily associate the elec-trical energy with the “kinetic energy” and the magneticenergy with the “potential energy” of the oscillator. Theinstantaneous, time-dependent energy in each element isderived from its current and voltage,

E(t) =∫ t

−∞V (t′)I(t′)dt′, (3)

where V (t′) and I(t′) denote the voltage and current ofthe capacitor or inductor.

To derive the classical Hamiltonian, we follow the stan-dard approach used in classical mechanics: the Lagrange-Hamilton formulation. Here, we represent the circuit el-ements in terms of one of its generalized circuit coor-dinates, charge or flux. In the following, we pick flux,defined as the time integral of the voltage

Φ(t) =∫ t

−∞V (t′)dt′. (4)

In this example, the voltage at the node is also the branchvoltage across the element. In this section, we will simplyrefer to these as node voltages and fluxes for convenience.For a more detailed discussion of nodes and branches inthis context, we refer the reader to Ref. 44.

Note that in the following, we could have exchangedour associations with kinetic energy (momentum coor-dinate) and potential energy (position coordinate), andinstead start with the charge variable Q(t), which is thetime integral of the current I(t).

By combining Eqs. (3) and (4), using the relations V =L dI/dt and I = C dV/dt, and applying the integrationby parts formula, we can write down energy terms for thecapacitor and inductor in terms of the node flux,

TC = 12CΦ2, (5)

UL = 12LΦ2. (6)

The Lagrangian is defined as the difference betweenthe kinetic and potential energy terms and can thus beexpressed in terms of Eqs. (5) and (6)

4

L = TC − UL = 12CΦ2 − 1

2LΦ2. (7)

From the Lagrangian in Eq. (7), we can further derivethe Hamiltonian using the Legendre transformation, forwhich we need to calculate the momentum conjugate tothe flux, which in this case is the charge on the capacitor

Q = ∂L∂Φ

= CΦ. (8)

The Hamiltonian of the system is now defined as

H = QΦ− L = Q2

2C + Φ2

2L ≡12CV

2 + 12LI

2, (9)

as one would expect for an electrical LC circuit. Notethat this Hamiltonian is analogous to that of a mechan-ical harmonic oscillator, with mass m = C, and reso-nant frequency ω = 1/

√LC, which expressed in posi-

tion, x, and momentum, p, coordinates takes the formH = p2/2m+mω2x2/2.

The Hamiltonian described above is classical. In orderto proceed to a quantum-mechanical description of thesystem, we need to promote the charge and flux coordi-nates to quantum operators. And, whereas the classicalcoordinates satisfy the Poisson bracket:

f, g = δf

δΦδg

δQ− δg

δΦδf

δQ(10)

→ Φ, Q = δΦδΦ

δQ

δQ− δQ

δΦδΦδQ

= 1− 0 = 1, (11)

the quantum operators similarly satisfy a commutationrelation:

[Φ, Q] = ΦQ− QΦ = i~, (12)

where the operators are indicated by hats. From thispoint forward, however, the hats on operators will beomitted for simplicity.

In a simple LC resonant circuit [Fig. 1(a)], both theinductor L and the capacitor C are linear circuit ele-ments. Defining the reduced flux φ ≡ 2πΦ/Φ0 and thereduced charge n = Q/2e, we can write down the follow-ing quantum-mechanical Hamiltonian for the circuit,

H = 4ECn2 + 12ELφ

2, (13)

where EC = e2/(2C) is the charging energy requiredto add each electron of the Cooper-pair to the islandand EL = (Φ0/2π)2/L is the inductive energy, whereΦ0 = h/(2e) is the superconducting magnetic flux quan-tum. Moreover, the quantum operator n is the ex-cess number of Cooper-pairs on the island, and φ – thereduced flux – is denoted the “gauge-invariant phase”

across the inductor. These two operators form a canon-ical conjugate pair, obeying the commutation relation[φ, n] = i. We note that the factor 4 in front of thecharging energy EC is solely a historical artifact, namely,that this energy scale was first defined for single-electronsystems and then adopted to two-electron Cooper-pairsystems.

The Hamiltonian in Eq. (13) is identical to the one de-scribing a particle in a one-dimensional quadratic poten-tial, a quantum harmonic oscillator (QHO). We can treatφ as the generalized position coordinate, so that the firstterm is the kinetic energy and the second term is the po-tential energy. We emphasize that the functional formof the potential energy influences the eigensolutions. Forexample, the fact that this term is quadratic (UL ∝ φ2)in Eq. (13) gives rise to the shape of the potential in Fig.1(b). The solution to this eigenvalue problem gives an in-finite series of eigenstates |k〉, (k = 0, 1, 2, . . .), whose cor-responding eigenenergies Ek are all equidistantly spaced,i.e. Ek+1−Ek = ~ωr, where ωr =

√8ELEC/~ = 1/

√LC

denotes the resonant frequency of the system, see Fig.1(b). We may represent these results in a more compactform (second quantization) for the quantum harmonicoscillator (QHO) Hamiltonian

H = ~ωr(a†a+ 1

2

), (14)

where a†(a) is the creation (annihilation) operator of asingle excitation of the resonator. The Hamiltonian inEq. (14) is written as an energy. It is, however, oftenpreferred to divide by ~ so that the expression has unitsof radian frequency, since we will later resonantly drivetransitions at a particular frequency or reference the rateat which two systems interact with one another. There-fore, from here on, ~ will be omitted.

The original charge number and phase operators canbe expressed as n = nzpf×i(a−a†) and φ = φzpf×(a+a†),where nzpf = [EL/(32EC)]1/4 and φzpf = (2EC/EL)1/4

are the zero-point fluctuations of the charge and phasevariables, respectively. Quantum mechanically, the quan-tum states are represented as wavefunctions that are gen-erally distributed over a range of values of n and φ and,consequently, the wavefunctions have non-zero standarddeviations. Such wavefunction distributions are referredto as “quantum fluctuations,” and they exist, even in theground state, where they are called “zero-point fluctua-tions”.

The linear characteristics of the QHO has a naturallimitation in its applications for processing quantum in-formation. Before the system can be used as a qubit, weneed to be able to define a computational subspace con-sisting of only two energy states (usually the two-lowestenergy eigenstates) in between which transitions can bedriven without also exciting other levels in the system.Since many gate operations, such as single-qubit gates(Sec. IV), depend on frequency selectivity, the equidis-tant level-spacing of the QHO, illustrated in Fig. 1(b),

5

0

4

1

2

3

5

Ener

gy [

]

Transmon

(d)

(c)

0

4

1

2

3

5

Ener

gy [

]

QHO

(a)

(b)

Co

mp

.

sub

spa

ce0- - /2 /2

Superconducting phase,0- - /2 /2

Superconducting phase,

CrLr CJ

LJ Cs

i

v+

-

FIG. 1. (a) Circuit for a parallel LC-oscillator (quantum har-monic oscillator, QHO), with inductance L in parallel withcapacitance, C. The superconducting phase on the island isdenoted φ, referencing ground as zero. (b) Energy potentialfor the QHO, where energy levels are equidistantly spaced~ωr apart. (c) Josephson qubit circuit, where the nonlinearinductance LJ (represented with the Josephson-subcircuit inthe dashed orange box) is shunted by a capacitance, Cs. (d)The Josephson inductance reshapes the quadratic energy po-tential (dashed red) into sinusoidal (solid blue), which yieldsnon-equidistant energy levels. This allows us to isolate thetwo lowest energy levels |0〉 and |1〉, forming a computationalsubspace with an energy separation ~ω01, which is differentthan ~ω12.

poses a practical limitation†.To mitigate the problem of unwanted dynamics in-

volving non-computational states, we need to add anhar-monicity (or nonlinearity) into our system. In short, werequire the transition frequencies ω0→1

q and ω1→2q be suffi-

ciently different to be individually adressable. In general,the larger the anharmonicity the better. In practise, theamount of anharmonicity sets a limit on how short thepulses used to drive the qubit can be. This is discussedin detail in Sec. IV D 3.

To introduce the nonlinearity required to modify theharmonic potential, we use the Josephson junction – anonlinear, dissipationless circuit element that forms thebackbone in superconducting circuits46,47. By replacingthe linear inductor of the QHO with a Josephson junc-tion, playing the role of a nonlinear inductor, we canmodify the functional form of the potential energy. Thepotential energy of the Josephson junction can be derivedfrom Eq. (3) and the two Josephson relations

†Even though linear resonant systems cannot be addressed properly,their long coherence times have proven them useful as quantum ac-cess memories for storing quantum information, where a nonlinearancilla system is used as a quantum controller for feeding and ex-tracting excitations to/from the resonant cavity modes45.

I = Ic sin(φ), V = ~2edφ

dt, (15)

resulting in a modified Hamiltonian

H = 4ECn2 − EJ cos(φ), (16)

where EC = e2/(2CΣ), CΣ = Cs + CJ is the total ca-pacitance, including both shunt capacitance Cs and theself-capacitance of the junction CJ , and EJ = IcΦ0/2π isthe Josephson energy, with Ic being the critical currentof the junction‡. After introducing the Josephson junc-tion in the circuit, the potential energy no longer takesa manifestly parabolic form (from which the harmonicspectrum originates), but rather features a cosinusoidalform, see the second term in Eq. (16), which makes theenergy spectrum non-degenerate. Therefore, the Joseph-son junction is the key ingredient that makes the oscilla-tor anharmonic and thus allows us to identify a uniquelyaddressable quantum two-level system, see Fig. 1(d).

Once the nonlinearity has been added, the system dy-namics is governed by the dominant energy in Eq. (16),reflected in the EJ/EC ratio. Over time, the super-conducting qubit community has converged towards cir-cuit designs with EJ EC . In the opposite case whenEJ ≤ EC , the qubit becomes highly sensitive to chargenoise, which has proven more challenging to mitigatethan flux noise, making it very hard to achieve high co-herence. Another motivation is that current technologiesallow for more flexibility in engineering the inductive (orpotential) part of the Hamiltonian. Therefore, workingin the EJ ≤ EC limit, makes the system more sensitiveto the change in the potential Hamiltonian. Therefore,we will focus here on the state-of-the-art qubit modalitiesthat fall in the regime EJ EC . For readers who areinterested in the physics in the EJ ≤ EC regime, suchas the earlier Cooper-pair box charge qubit, we refer toRefs. 48–51.

To access the EJ EC regime, one preferred approachis to make the charging EC small by shunting the junctionwith a large capacitor, Cs CJ , effectively making thequbit less sensitive to charge noise – a circuit commonlyknown as the transmon qubit52. In this limit, the super-conducting phase φ is a good quantum number, i.e. thespread (or quantum fluctuation) of φ values representedby the quantum wavefunction is small. The low-energyeigenstates are therefore, to a good approximation, local-ized states in the potential well, see Fig. 1(d). We maygain more insight by expanding the potential term of Eq.(16) into a power series (since φ is small), that is

‡The critical current is the maximum supercurrent that the junctioncan support before it switches to the resistive state with non-zerovoltage.

6

EJ cos(φ) = 12EJφ

2 − 124EJφ

4 +O(φ6). (17)

The leading quadratic term in Eq. (17) alone willresult in a QHO, recall Eq. (13). The second term,however, is quartic which modifies the eigensolution anddisrupts the otherwise harmonic energy structure. Notethat, the negative coefficient of the quartic term indicatesthat the anharmonicity α = ω1→2

q −ω0→1q is negative and

its limit in magnitude thus cannot be made arbitrarilylarge. For the case of the transmon, α = −EC is usuallydesigned to be 100 − 300 MHz, as required to maintaina desirable qubit frequency ωq = (

√8EJEC − EC)/~ =

3−6 GHz, while keeping an energy ratio sufficiently large(EJ/EC ≥ 50) to suppress charge sensitivity52. Fortu-nately, the charge sensitivity is exponentially suppressedfor increased EJ/EC , while the reduction in anharmonic-ity only scales as a weak power law, leading to a workabledevice.

Including terms up to fourth order and using the QHOeigenbases, the system Hamiltonian resembles that of aDuffing oscillator

H = ωqa†a+ α

2 a†a†aa. (18)

Since |α| ωq, we can see that the transmon qubit isbasically a weakly anharmonic oscillator (AHO). If exci-tation to higher non-computational states is suppressedover any gate operations, either due to a large enough |α|or due to robust control techniques such as the DRAGpulse, see Sec. IV D 3, we may effectively treat the AHOas a quantum two-level system, simplifying the Hamilto-nian to

H = ωqσz2 , (19)

where σz is the Pauli-z operator. However, one shouldalways keep in mind that the higher levels physicallyexist53. Their influence on system dynamics should betaken into account when designing the system and itscontrol processes. In fact, there are many cases wherethe higher levels have proven useful to implement moreefficient gate operations54.

In addition to reducing the charge dispersion, the useof a large shunt capacitor also enables us to engineerthe electric field distribution of the quantum system, andthus the participation of surface loss mechanisms. In thedevelopment of the 3D transmon55, e.g. a 2D transmoncoupled to a 3D cavity, it was demonstrated that by mak-ing the gap between the two lateral capacitor plates large(compared to the film thickness) the coherence time in-creases since a smaller portion of the electric field in-teracts with the lossy interfaces, e.g. metal-substrateand substrate-vacuum interfaces, which has been stud-ied extensively56–61.

B. Qubit Hamiltonian engineering

1. Tunable qubit: split transmon

To implement fast gate operations with high-fidelity,as needed to implement quantum logic, many (thoughnot all63) of the quantum processor architectures imple-mented today feature tunable qubit frequencies64–67. Forinstance, in some cases, we need to bring two qubits intoresonance to exchange (swap) energy, while we also needthe capability of separating them during idling periodsto minimize their interactions. To do this, we need anexternal parameter which allows us to access one of thedegrees of freedom of the system in a controllable fashion.

One widely-used technique is to replace the singleJosephson junction with a loop interupted by two iden-tical junctions – forming a dc superconducting quantuminterference device (dc-SQUID)68. Due to the interfer-ence between the two arms of the SQUID, the effectivecritical current of the two parallel junctions can be de-creased by applying a magnetic flux threading the loop,see Fig. 2(a). Due to the fluxoid quantization condition,the algebraic sum of branch flux of all of the inductiveelements along the loop plus the externally applied fluxequal an integer number of superconducting flux quanta,that is

ϕ1 − ϕ2 + 2ϕe = 2πk, (20)

where ϕe = πΦext/Φ0. Using this condition, we can elim-inate one degree of freedom and treat the SQUID-loopas a single junction, but with the important modificationthat EJ is tunable (via the SQUID critical current) bymeans of the external flux Φext. The effective Hamilto-nian of the so-called split transmon (ignoring the con-stant) is

H = 4ECn2 − 2EJ |cos (ϕe)|︸ ︷︷ ︸E′J

(ϕe)

cos(φ). (21)

We can see that Eq. (21) is analogous to Eq. (16), withEJ replaced by E′J(ϕe) = 2EJ |cos (ϕe)|. The magnitudeof the net, effective Josephson energy E′J has a periodof Φ0 in applied flux and spans from 0 to its maximumvalue 2EJ . Therefore, the qubit frequency can be tunedperiodically with Φext, see Fig. 2(b).

While the split transmon enables frequency tunabil-ity by the externally applied magnetic field, it also in-troduces sensitivity to random flux fluctuations, knownas flux noise. At any working point, the slope of thequbit spectrum, ∂ωq/∂Φext, indicates to first order howstrongly this flux noise affects the qubit frequency. Thesensitivity is generally non-zero, except at multiples ofthe flux quantum, Φext = kΦ0, where k is an integer,where ∂ωq/∂Φext = 0.

7

(c)

(d)

(a)

(b)8

6

4

2

0

8

6

4

2

0

(g)

(h)

(e)

(f)8

6

4

2

8

6

4

2

00- - /2 /2

Asymmetric transmon C-shunted Flux qubit C-shunted FluxoniumSymmetric transmon

00- - /2 /20- - /2 /20- - /2 /2

IcIc Ic

φ1 φ2φ1 φ2φ1

φ2

φ1φ2

φNCJ

Ec /h = 0.3 GHzEJ /h = 15 GHz

FIG. 2. Modular qubit circuit representations for capacitively shunted qubit modalities (orange box Fig. 1c) and correspondingqubit transition frequencies for the two lowest energy states as a function of applied magnetic flux in units of Φ0. (a-b)Symmetric transmon qubit, with Josephson energy EJ are shunted with a capacitor yielding a charging energy EC . (c-d)Asymmetric transmon qubit, with junction asymmetry γ = EJ2/EJ1 = 2.5. (e-f) Capacitively shunted flux qubit, where asmall principle junction (red) is shunted with two larger junctions (orange). Parameters are the same as Yan et al.62. (g-h)C-shunted fluxonium qubit, where the small junction is inductively shunted with a large array of N junctions.

One recent development has focused on reducing thequbit sensitivity to flux noise, while maintaining suf-ficient tunability to operate our quantum gates. Theidea is to make the two junctions in the split transmonasymmetric69, see Fig. 2(c). This yields the followingHamiltonian

H = 4ECn2−EJΣ

√cos2(ϕe) + d2 sin2(ϕe)︸ ︷︷ ︸

E′J

(ϕe)

cos(φ), (22)

where EJΣ = EJ1 + EJ2 and d = (γ − 1)/(γ + 1) isthe junction asymmetry parameter, with γ = EJ2/EJ1.Again, we can treat the two junctions as a single-junctiontransmon, with an effective Josephson energy E′J(ϕe). Inparticular, we can recognize the two special cases; ford = 0, the Hamiltonian in Eq. (22) reduces to the sym-metric case with E′J(ϕe) = EJΣ |cos(ϕe)|, as in Eq. (21)with EJΣ = 2EJ . In the other limit, when |d| → 1,E′J(ϕe) → EJΣ and the flux-tunability of the Josephsonenergy vanishes, which is equivalent to the single junctioncase, recall Eq. (16).

From the discussion above we see that going from sym-metric to asymmetric transmons does not change the cir-cuit topology. This seemingly trivial modification, how-ever, has profound impact for practical applications. Aswe can see from the qubit spectra, Fig. 2(d), the fluxsensitivity is suppressed across the entire tunable fre-quency range. For example, the performance of the cross-

resonance gate is optimized with certain frequency de-tuning between two qubits70. Therefore, by using anasymmetric transmon, a small frequency-tuning rangeis introduced that is sufficient to compensate for fabri-cation variations, without introducing unnecessary largesusceptibility to flux noise and thus maintaining high co-herence. For another example, a surface code schemebased on the adiabatic CPHASE-gate requires specific fre-quency configuration among qubits in order to avoid fre-quency crowding issues, and asymmetric transmons fitwell with its well-defined frequency range71. In general,as the quantum processors scale up and fabrication im-proves, asymmetric transmons are likely to be found inwider applications in the future.

2. Towards larger anharmonicity: flux qubit and fluxonium

We see that split transmon qubits, be it symmetric ornot, still share the same topology as the single junctionversion, yielding a sinusoidal potential. Therefore, thedegree to which the properties of these qubits can be en-gineered has not fundamentally changed. In particular,the limited anharmonicity in transmon-type qubits in-trinsically causes significant residual excitation to higher-energy states, undermining performance of gate opera-tions. To go beyond this, it is necessary to introduceadditional complexity into the circuit.

One outstanding development in this regard is the in-

8

vention of the flux qubit72,73, where the qubit loop is in-terrupted by three (or four) junctions, see Fig. 2(e). Onone branch is one smaller junction; on the other branchare two identical junctions, both a factor γ larger in sizecompared to the small junction. The addition of onemore junction as compared to the split transmon is non-trivial, as it changes the circuit topology and reshapesthe potential energy profile.

Each junction is associated with a phase variable, andthe fluxoid quantization condition again allows us toeliminate one degree of freedom. Consequently, we havea two-dimensional potential landscape, which in compar-ison to the simpler topology of the transmon, compli-cates the problem both conceptually and computation-ally. Fortunately, under the assumed setting that the ar-ray junctions are larger in size (γ > 1), it is usually a goodapproximation to treat the problem as a particle movingin a quasi-1D potential, which also helps us gain moreinsight and intuition about the system and draw qualita-tive conclusions. The Hamiltonian under this quasi-1Dapproximation reads,

H ≈ 4ECn2 − EJ cos(2φ+ ϕe)− 2γEJ cos(φ). (23)

Note that the phase variable in Eq. (23) is the sumof the branch phases across the two array junctions, φ =(ϕ1 + ϕ2)/2, assuming the same current direction acrossϕ1 and ϕ2. The external magnetic flux is denoted ϕe =2πΦext/Φ0. The second term in Eq. (23) is contributedby the small junction with Josephson energy EJ , whereasthe third term takes into account the two array junctions,together with Josephson energy 2γEJ . Clearly, the sumof these two terms no longer has the characteristics of asimple cosinusoid, and the final potential profile as wellas the corresponding eigenstates depends on both theexternal flux ϕe and the junction area ratio γ.

The most common working point for this system iswhen ϕe = π + 2πk, where k is an integer – that iswhen half a superconducting flux quantum threads thequbit loop. At this flux bias point, the qubit spectrumreaches its minimum, and the qubit frequency is first-order insensitive to flux noise, see Fig. 2(f). This pointis often referred to as the flux degeneracy point, whereflux qubits tend to have the optimal coherence time.

At this operation point, the potential energy may as-sume a single-well (γ ≥ 2) or a double-well (γ < 2)profile. The single-well case shares some simularitieswith the transmon qubit, where the quadratic and quar-tic terms of the Hamiltonian determines the harmonic-ity and anharmonicity, respectively. The capacitively-shunted flux qubit (CSFQ)62,74 was explored in thisregime, demonstrating long coherence and decently highanharmonicity. Note that as opposed to the transmonqubit, the anharmonicity of the CSFQ is positive (α > 0).While the improvement in anharmonicity can be associ-ated with reshaping the energy potential, the improvedcoherence over the first flux qubits can be attributed to

the introduction of the capacitive shunt, similar to themodified Cooper-pair box leading to the transmon qubit.

The double-well case obtained for γ < 2 was demon-strated and investigated much earlier72,73. The intuitivepicture based on circulating current states – so it getsthe name persisting-current flux qubit (PCFQ) – givesa satisfying physical description of the qubit degrees offreedom. However, from the perspective of a quantumengineer, the qubit properties are of more interest, evenif sometimes we may lose physical intuition about thesystem in certain regimes; such as when γ ≈ 2 and thereare no clear circulating current states. The most impor-tant feature of the PCFQ is that its anharmonicity canbe much greater than the transmon and CSFQ and thetransition matrix elements |〈1|n|0〉|, |〈1|φ|0〉| become con-siderably smaller given equivalent EJ/EC . Therefore, alonger relaxation time can be expected. These featureshave been demonstrated even more prominently in itsclose relative, the fluxonium qubit75.

The flux qubit is a striking example that illustrateshow one dramatically can engineer the qubit propertiesthrough the choice of various circuit parameters. The in-troduction of array junctions and consequent biharmonicprofile generates rich dynamics as well as broad applica-tions. An extention of this idea is the fluxonium qubit,which generated substantial interest recently, due partlyto its capability of engineering the transition matrix el-ements to achieve millisecond T1 time, and due partlyto the invention of novel gate schemes applicable to suchwell-protected qubits76,77.

Compared to flux qubits, which usually contain twoor three array junctions78, the number of array junctionsin the fluxonium qubit is dramatically increased75,79, insome cases, to the order of 100, see Fig. 2(g). Followingthe same quasi-1D approximation as for the flux qubit,the last term in Eq. (23) becomes −NγEJ cos(φ/N),where N denotes the number of array junctions. Forlarge N , the argument in the cosine term φ/N becomessufficiently small that a second order expansion is a goodapproximation. This results in the fluxonium Hamilto-nian,

H ≈ 4ECn2 − EJ cos(φ+ ϕe) + 12ELφ

2, (24)

where EL = (γ/N)EJ is the inductive energy of theeffective inductance contributed by the junction ar-ray – often known as superinductance due to its largevalue79–81. Therefore, we can treat the potential energyas a quadratic term modulated by a sinusoidal term, sim-ilar to that of an rf-SQUID type flux qubit82. However,the kinetic inductance of the Josephson junction array isin general much larger than the geometric inductance ofthe wire in an rf-SQUID.

Depending on the relative magnitude of EJ and EL,the fluxonium system could involve plasmon states (inthe same well) and fluxon states (in different wells).There are a variety of schemes to utilize them for quan-tum information processing. Generally, the spectrum of

9

the transition between the lowest energy states is similarto that of the flux qubit, see Fig. 2(h). Both long coher-ence and high anharmonicity can be expected at the fluxsweet spot.

Lastly, we want to point out a further extension – the0−π qubit – which has even stronger topological protec-tion from noise83,84. However, the strongly suppressedsensitivity to external fluctuations also makes it hard tomanipulate.

C. Interaction Hamiltonian engineering

To generate entanglement between individual quantumsystems – it is necessary to engineer an interaction Hamil-tonian that connects degrees of freedom in those indi-vidual systems. In this section, we discuss the physicalcoupling mechanism and its representation in the qubiteigenbasis. The use of coupling to form 2-qubit gates isdiscussed in Sec. IV.

1. Physical coupling: capacitive and inductive

The Hamiltonian of two coupled systems takes ageneric form

H = H1 +H2 +Hint, (25)

where H1 and H2 denote the Hamiltonians of the individ-ual quantum systems, which could be any combination ofthe qubit circuits mentioned in Sec. II A and II B. Thelast term, Hint, is the interaction Hamiltonian, whichcouples variables of both systems. In superconductingcircuits, the physical form of the coupling energy is eitheran electric or magnetic field (or a combination thereof).

To achieve capacitive coupling, a capacitor is placedbetween the voltage nodes of the two participating cir-cuits, yielding an interaction Hamiltonian Hint of theform

Hint = CgV1V2, (26)

where Cg is the coupling capacitance and V1(V2) is thevoltage operator of the corresponding voltage node beingconnected. Fig. 3(a) illustrates a realistic example of adirect capacitive coupling between the top nodes of twotransmon qubits. Circuit quantization in the limit ofCg C1, C2 yields

H =∑i=1,2

[4EC,in2

i − EJ,i cos(φi)]

+4e2 CgC1C2

n1n2, (27)

Cg

C1 C2IC2IC1

Cg1

C1 C2IC2IC1

Cg2

CrLr

L1 IC2IC1 L2

M12

Φe1 Φe2

L1IC1 L2

M1C

Φe1 ΦeCIC2

Φe2

M2C

(a) Direct capacitive coupling

(b) Capacitive coupling via coupler

(c) Direct inductive coupling

(d) Inductive coupling via coupler

V1 V2

I1 I2

ICC

gr1 gr2

g12

FIG. 3. Schematic of capacitive and inductive couplingschemes between two superconducting qubits, labeled 1 and2. (a) Direct capacitive coupling, where the voltage nodesof two qubits V1 and V2 are connected by a capacitance Cg.(b) Capacitive coupling via a coupler in form of a linear res-onator. (c) Direct inductive coupling, where the two qubitsare coupled via mutual inductance, M12. (d) Inductive cou-pling via mutual inductances M1C and M2C to a frequency-tunable coupler.

where the expressions in brackets are the two Hamiltoni-ans of the individual qubits, [see Eq. (16)], and we takeVi = (2e/Ci)ni in Eq. (26). From Eq. (27), we see thatthe coupling energy depends on the coupling capacitanceas well as the matrix elements of the voltage operators.The dependencies are bilinear in the perturbative limit(Cg C1, C2).

To implement the coupling capacitance, one only needbring the edges of the capacitor pads into close proxim-ity, as has been demonstrated in state-of-the-art planardesigns85. The coupling capacitance is determined bythe planar capacitor geometry as well as the surround-ing environment, such as the dielectric constant of thesubstrate and the ground plane proximity.

10

In the case of inductive coupling, a mutual inductanceshared by two loops is the coupling mechanism, yieldingan interaction Hamiltonian of the form

Hint = M12I1I2, (28)

where M12 denotes the mutual inductance and I1(I2) isthe current-operator of the loop current. A typical exam-ple is two closely positioned (rf-SQUID type) flux qubits,as illustrated in Fig. 3(c). The system Hamiltonian canbe expressed as

H =∑i=1,2

[4EC,in2

i + 12EL,iφ

2i − EJ,i cos(φi)

]

+M12Ic1 sin(φ1)Ic2 sin(φ2), (29)

where the individual qubit Hamiltonians are identical tothat of the fluxonium in Eq. (24), and the current op-erators, Ii = Ici sin(φi) with i ∈ 1, 2, is the familiar dc-Josephson relation for each junction, see Eq. (15). Inthis case, the strength of the inductive coupling energydepends on the mutual inductance as well as the matrixelement of the current operators.

To realize a mutual inductance, two looped circuitsare brought into close proximity to one another, or, tomake it stronger, overlap with each other86, and evenmay share the same wire or Josephson junction induc-tor87–90. In the case of a Josephson junction, and forcertain metals, the inductance is dominated by kineticinductance contributions, rather than solely geometricinductance91,92. Kinetic inductance arises from the me-chanical, inertial mass of the charge carriers, but is onlypractically witnessed in very high-conductance materi-als like superconductors. A primary feature of kineticinductance is that its values can vastly exceed those ofconventional geometric inductances, which are generallylimited by electromagnetic considerations79.

2. Coupling axis: transverse and longitudinal

Regardless of its physical realization, the effect of acoupling on system dynamics is determined by its formas represented in the eigenbasis of the individual systems.That is, how Hint appears in the representation spannedby the eigenbasis of H1 ⊗H2.

Let us start with the previous example of two capaci-tively coupled transmon qubits [Fig. 3(a)]. Using secondquantization, the system Hamiltonian in Eq. (27) can beexpressed as

H =∑i∈1,2

[ωia†iai + αi

2 a†ia†iaiai

]

−g(a1 − a†1

)(a2 − a†2

), (30)

where the expression within brackets represent the Duff-ing oscillator Hamiltonian for the qubits and g is thecoupling energy. Since we define V ∝ n ∝ i(a− a†), andconsequently I ∝ φ ∝ (a + a†), the original n1n2-termbecomes what is shown in Eq. (30). Such a coupling iscalled transverse, because the coupling Hamiltonian hasnon-zero matrix elements only at off-diagonal positionswith respect to both oscillators, i.e. i〈k|ai − a†i |k〉i = 0for any integer k and for i ∈ 1, 2 and in this casei〈k ± 1|ai − a†i |k〉i 6= 0.

If we can ignore higher energy levels (k ≥ 2) eitherbecause of sufficient anharmonicity or through carefulcontrol protocols that ensure these levels never have in-fluence, we may truncate the Hamiltonian in Eq. (30)to

H =∑i∈1,2

12ωiσz,i + gσy,1σy,2. (31)

This is a Hamiltonian of two spins, coupled by an ex-change interaction. As we will see in Sec. IV D 1, sucha Hamiltonian is most commonly used in contemporaryimplementations and can generate various types of two-qubit entangling gates. Note that, more often, we seethat the interaction term is expressed in σxσx instead ofσyσy. The choice in the context here is arbitrary and doesnot change the dynamics. However, when both capaci-tive and inductive couplings are present in the system,both σxσx and σyσy may be needed. In this case, thevoltage operator V ∝ i(a− a†) (reduced to σy after two-level approximation in the lab frame) is transversal to thecurrent operator I ∝ (a+a†) (reduced to σx) and both ofthem may be transverse to the qubit. A similar exampleis demonstrated between a qubit and a resonator by Luet al.93

Transverse coupling can be engineered between a qubitand a harmonic oscillator, see Fig. 3(b). In this case, theHamiltonian becomes

H = 12ωqσz + ωra

†a+ g(σ+a+ σ−a†), (32)

where ωq and ωr denote the qubit and resonator fre-quencies, and σ+ = |0〉〈1| and σ− = |1〉〈0| describesthe processes of exciting and de-exciting the qubit, re-spectively. Here, we have assumed that the coupling isin the dispersive limit, i.e. g ωq, ωr, hence ignoringthe double (de)excitation terms proportional to σ+a

† andσ−a, which under typical operation regimes oscillate suf-ficiently fast to average to zero. The Hamiltonian in Eq.(32), is the standard model used for describing how a two-level atom interacts with a resonant cavity that housesit. Such a structure is also known as cavity quantum

11

electrodynamics (cQED), and it is extended to the cir-cuit version here. It has many useful applications in su-perconducting quantum information architectures, suchas high-fidelity readout94, see Sec. V, cavity buses95,quantum memory96,97, quantum computation with catstates98–100, etc.

Here, we briefly mention the use of a cavity or res-onator to mediate coupling between two qubits, whichmay be physically well-separated (≈ 1 cm). Since mostsuperconducting resonators are in the GHz frequencyrange, they can be made much longer than any dimen-sion of a qubit circuit (≈ 1 mm). One can use such aresonator to mediate coupling between two or more oth-erwise non-interacting qubits. An example is shown inFig. 3(b), where two transmon qubits are both capac-itively coupled to the center resonator. The two-levelsystem Hamiltonian is:

H =∑i=1,2

(ωia†iai + αi

2 a†ia†iaiai

)+ ωra

†rar

+g1r

(a†1ar + a1a

†r

)+ g2r

(a†2ar + a2a

†r

). (33)

It can be shown that in the dispersive limit, i.e. gir |ωi−ωr|, the resonator can – after proper transformationand approximation – be treated as an isolated system,and the composite system simplified to two transverselycoupled qubits, see Eq. (31).

We now turn to the previous example of two induc-tively coupled flux qubits, see Fig. 3(c). Assume thatthe double-well potential [Fig. 2(g)] has a relativelyhigh inter-well barrier, which leads to an exponentiallysmall qubit transition frequency at the energy degener-acy point, (Φe = π). Around this degeneracy point, theoff-diagonal matrix element of sin(φ) is zero, i.e. theground and excited states are localized in different wellsand 〈1| sin(φ)|1〉− 〈0| sin(φ)|0〉 6= 0. We can then rewritethe Hamiltonian in Eq. (29) as

H =∑i=1,2

12ωiσzi + gσz1σz2. (34)

Now, the coupling axis is the same as the qubit quan-tization axes and therefore termed longitudinal coupling.Note, however, that the physical σxσx and σzσz couplingscan change in the qubit frame.

Longitudinal coupling is an important type of inter-action, because it can generate entanglement withoutenergy exchange. Moreover, it is found a necessary in-gredient in the application of quantum annealing, wherecertain hard combinatorial optimization problems can bemodeled by the Ising Hamiltonian in Eq. (34) and findingits ground state would solve this problem.

An intermediate qubit mode may also be used as a cou-pler in the longitudinal case. In Fig. 3(d), an additionalrf-SQUID is used to mediate the coupling. The coupling

strength can be tuned by the flux bias of the couplerSQUID101. Note that a tunable coupler may also be re-alized in a structure with capacitive couplings63. A tun-able coupler is useful because it provides a wide range ofcoupling strengths102, a high on-off ratio103 for reducinggate error-rates, and more ways of achieving high-fidelityentangling gates67,104–106. The trade-off is an additionalcontrol line.

In addition to the pure transversal and longitudinalqubit-qubit interactions, there are also examples of mixedtypes of interaction Hamiltonians107

H = 12ωqσz + ωra

†a+ gσz(a+ a†), (35)

which are longitudinal with respect to a qubit, but trans-verse with respect to a harmonic oscillator in a qubit-resonator system. Such a model is called longitudinalbut one should note that it is only longitudinal to oneparticipating system. It is hard to engineer physicallylongitudinal coupling with respect to a harmonic oscil-lator, since either the E-field (V ) or the B-field (I) istransverse with respect to the eigen field of the harmonicoscillator. Note, however, that a transversal model suchas in Eq. (32) may be transformed into a longitudinalone in certain operating regimes, see Sec. V.

In some applications, such as for quantum annealing,both longitudinal and transverse couplings are desired(σzσz coupling for mapping the problem and σxσx cou-pling for enhancing the annealing performance) and re-quire independent control.

III. NOISE, DECOHERENCE, AND ERRORMITIGATION

Random, uncontrollable physical processes in the qubitcontrol and measurement equipment or in the local envi-ronment surrounding the quantum processor are sourcesof noise that lead to decoherence and reduce the opera-tional fidelity of the qubits. In this section, we introducethe basics of noise leading to decoherence in supercon-ducting circuits, and we discuss coherent control methodsto mitigate certain types of noise.

A. Types of noise

In a closed system, the dynamical evolution of a qubitstate is deterministic. That is, if we know the startingstate of the qubit and its Hamiltonian, then we can pre-dict the state of the qubit at any time in the future.However, in open systems, the situation changes. Thequbit now interacts with uncontrolled degrees of freedomin its environment, which we refer to as fluctuations ornoise. In the presence of noise, as time progresses, thequbit state looks less and less like the state we wouldhave predicted and, eventually, the state is lost. There

12

are many different sources of noise that affect quantumsystems, and they can be categorized into two primarytypes: systematic noise and stochastic noise.

1. Systematic noise

Systematic noise arises from a process that is trace-able to a fixed control or readout error. For example, weapply a microwave pulse to the qubit that we believewill impart a 180-degree rotation. However, the con-trol field is not tuned properly and, rather than rotatingthe qubit 180 degrees, the pulse slightly over-rotates orunder-rotates the qubit by a fixed amount. The underly-ing error is systematic, and it therefore leads to the samerotation error each time it is applied. However, whensuch erroneous pulses are used in practice in a varietyof control sequences, the observed results may appearto be influenced by random noise. This is because thepulse is generally not applied in the same way for eachexperiment: it could be applied a different number oftimes, interspersed with different pulses in different or-ders, and therefore generally differs from experiment toexperiment. However, once systematic errors are iden-tified, they can generally be corrected through propercalibration or the use of improved hardware.

2. Stochastic noise

The second type of noise is stochastic noise, arisingfrom random fluctuations of parameters that are coupledto our qubit108. For example, thermal noise of a 50Ω re-sistor in the control lines leading to the qubit will havevoltage and current fluctuations – Johnson noise – with anoise power that is proportional to both temperature andbandwidth. Or, the oscillator that provides the carrierfor a qubit control pulse may have amplitude or phasefluctuations. Additionally, randomly fluctuating electricand magnetic fields in the local qubit environment – e.g.,on the metal surface, on the substrate surface, at themetal-substrate interface, or inside the substrate – cancouple to the qubit. This creates unknown and uncon-trolled fluctuations of one or more qubit parameters, andthis leads to qubit decoherence.

3. Noise strength and qubit susceptibility

The degree to which a qubit is affected by noise isrelated to the amount of noise impinging on the qubit,and the qubit’s susceptibility to that noise. The formeris often a question of materials science and fabrication;that is, can we make devices with lower levels of noise.Or, it may be related to the quality of the control elec-tronics and cryogenic engineering to limit the levels ofnoise on the control lines that necessarily connect to thequbits to control them. The latter – qubit susceptibility

– is a question of qubit design. Qubits can be designed totrade off sensitivity to one type of noise at the expense ofincreased sensitivity to other types of noise. Thus, mate-rials science, fabrication engineering, electronics design,cryogenic engineering, and qubit design all play a rolein creating devices with high coherence. In general, oneshould strive to eliminate the sources of noise, and thendesign qubits that are insensitive to the residual noise.

The qubit response to noise depends on how the noisecouples to it – either through a longitudinal or a trans-verse coupling as referenced to the qubit quantizationaxis. This can be visualized using a Bloch Sphere pictureof the qubit state, as illustrated in Fig. 4 and discussedin detail in Section III B.

B. Modeling noise and decoherence

1. Bloch sphere representation

The Bloch sphere is a unit sphere used to represent thequantum state of a two-level system (qubit). Fig. 4(a)shows a Bloch sphere with a Bloch vector representingthe state |ψ〉 = α|0〉 + β|1〉. If we visualize the Blochsphere as the planet Earth, then by convention, the northpole represents state |0〉 and the south pole state |1〉. Forpure quantum states such as |ψ〉, the Bloch vector is ofunit length, |α|2 + |β|2 = 1, connecting the center of thesphere to any point on its surface.

The z-axis connects the north and south poles. Itis called the longitudinal axis, since it represents thequbit quantization axis for the states |0〉 and |1〉 in thequbit eigenbasis. In turn, the x-y plane is the trans-verse plane with transverse axes x and y. In thisCartesian coordinate system, the unit Bloch vector ~a =(sin θ cosφ, sin θ sinφ, cos θ) is represented using the polarangle 0 ≤ θ ≤ π and the azimuthal angle 0 ≤ φ < 2π, asillustrated in Fig. 4 (a). Following our convention, state|0〉 at the north pole is associated with +1, and state |1〉(the south pole) with −1. We can similarly represent thequantum state using the angles θ and φ,

|ψ〉 = α|0〉+ β|1〉 = cos θ2 |0〉+ eiφ sin θ2 |1〉. (36)

The Bloch vector is stationary on the Bloch sphere inthe rotating frame picture. If state |1〉 has a higher en-ergy than state |0〉 (as it generally does in superconduct-ing qubits), then in a stationary frame, the Bloch vectorwould precess around the z-axis at the qubit frequency(E1−E0)/~. Without loss of generality (and much easierto visualize), we instead choose to view the Bloch spherein a reference frame where the x and y-axes also rotatearound the z-axis at the qubit frequency. In this rotatingframe, the Bloch vector appears stationary as written inEq. (36). The rotating frame will be described in detailin Section IV D 1 in the context of single-qubit gates.

For completeness, we note that the density matrix ρ =

13

(a) (b) (c)Bloch sphere Pure dephasingLongitudinal relaxation Transverse relaxation(d)

|0〉

|1〉

x

y

z

|Ψ〉 = α|0〉 + β|1〉 |0〉

|1〉

x

y

z|0〉

|1〉

x

y

z(Longitudinal)

(Transverse)

(Transverse)

RelaxationΓ

DecoherenceΓθ

φ

Transversenoise

Transversenoise

|0〉

|1〉

x

y

z

Pure Dephasing

Γφ

Γφ

Longitudinalnoise

ExcitationΓ1 1 2

Γφ

Γ1

FIG. 4. Transverse and longitudinal noise represented on the Bloch sphere. (a) Bloch sphere representation of the quantumstate |ψ〉 = α|0〉 + β|1〉. The qubit quantization axis – the z axis – is longitudinal in the qubit frame, corresponding to σzterms in the qubit Hamiltonian. The x-y plane is transverse in the qubit frame, corresponding to σx and σy terms in thequbit Hamiltonian. (b) Longitudinal relaxation results from energy exchange between the qubit and its environment, dueto transverse noise that couples to the qubit in the x − y plane and drives transitions |0〉 ↔ |1〉. A qubit in state |1〉 emitsenergy to the environment and relaxes to |0〉 with a rate Γ1↓ (blue arched arrow). Similarly, a qubit in state |0〉 absorbs energyfrom the environment, exciting it to |1〉 with a rate Γ1↑ (orange arched arrow). In the typical operating regime kBT ~ωq,the up-rate is suppressed, leading to the overall decay rate Γ1 ≈ Γ1↓. (c) Pure dephasing in the transverse plane arises fromlongitudinal noise along the z axis that fluctuates the qubit frequency. A Bloch vector along the x-axis will diffuse clockwise orcounterclockwise around the equator due to the stochastic frequency fluctuations, depolarizing the azimuthal phase with a rateΓφ. (d) Transverse relaxation results in a loss of coherence at a rate Γ2 = Γ1/2+Γφ, due to a combination of energy relaxationand pure dephasing. Pure dephasing leads to decoherence of the quantum state (1/

√2)(|0〉 + |1〉), initially pointed along the

x-axis. Additionally, the excited state component of the superposition state may relax to the ground state, a phase-breakingprocess that loses the orientation of the vector in the x-y plane.

|ψ〉〈ψ| for a pure state |ψ〉 is equivalently

ρ = 12(I + ~a · ~σ) = 1

2

(1 + cos θ e−iφ sin θeiφ sin θ 1 + sin θ

)(37)

=(

cos2 θ2 e−iφ cos θ2 sin θ

2eiφ cos θ2 sin θ

2 sin2 θ2

)(38)

=(|α|2 αβ∗

α∗β |β|2)

(39)

where I is the identity matrix, and ~σ = [σx, σy, σz] is avector of Pauli matrices. If the Bloch vector ~a is a unitvector, then ρ represents a pure state ψ and Tr(ρ2) = 1.More generally, the Bloch sphere can be used to representmixed states, for which |~a| < 1; in this case, the Blochvector terminates at points inside the unit sphere, and0 ≤ Tr(ρ2) < 1. To summarize, the surface of the unitsphere represents pure states, and its interior representsmixed states6.

2. Bloch-Redfield model of decoherence

Within the standard Bloch-Redfield109–111 picture oftwo-level system dynamics, noise sources weakly coupledto the qubits have short correlation times with respectto the system dynamics. In this case, the relaxation pro-

cesses are characterized by two rates (see Fig. 4):

longitudinal relaxation rate: Γ1 ≡1T1

(40)

transverse relaxation rate: Γ2 ≡1T2

= Γ1

2 + Γϕ(41)

which contains the pure dephasing rate Γϕ. We notethat the definition of Γ2 as a sum of rates presumes thatthe individual decay functions are exponential, which oc-curs for Lorentzian noise spectra (centered at ω = 0)such as white noise (short correlation times) with a high-frequency cutoff.

The impact of noise on the qubit can be visualized onthe Bloch sphere in Fig. 4(a). For an initial state (t = 0)

|ψ〉 = α|0〉+ β|1〉, (42)

the Bloch-Redfield density matrix ρBR for the qubit iswritten112,113,

ρBR =(

1 + (|α|2 − 1)e−Γ1t αβ∗eiδωte−Γ2t

α∗βe−iδωte−Γ2t |β|2e−Γ1t

). (43)

There are a few important distinctions between Eq. (43)and Eq. (39), which we list here and then describe inmore detail in subsequent sections.

• First, we have introduced the longitudinal decayfunction exp(−Γ1t), which accounts for longitudi-nal relaxation of the qubit.

• Second, we introduced the transverse decay func-tion exp(−Γ2t), which accounts for transverse de-cay of the qubit.

14

• Third, we have introduced an explicit phase ac-crual exp(iδωt), where δω = ωq − ωd, which gen-eralizes the Bloch sphere picture to account forcases where the qubit frequency ωq differs from therotating-frame frequency ωd, as we will see laterwhen discussing measurements of T2 using Ramseyinterferometry114,115, and in Section IV D 1 in thecontext of single-qubit gates.

• And, fourth, we have constructed the matrix suchthat for t (T1, T2), the upper-left matrix ele-ment will approach unit value, indicating that allpopulation relaxes to the ground state, while theother three matrix elements decay to zero. This isrelated to the assumption that the environmentaltemperature is low enough that thermal excitationsof the qubit from the ground to excited state rarelyoccur.

Longitudinal relaxationThe longitudinal relaxation rate Γ1 describes depolar-

ization along the qubit quantization axis, often referredto as “energy decay” or “energy relaxation.” In this lan-guage, a qubit with polarization p = 1 is entirely in theground state (|0〉) at the north pole, p = −1 is entirelyin the excited state (|1〉) at the south pole, and p = 0 isa completely depolarized mixed state at the center of theBloch sphere.

As illustrated in Fig. 4(b), longitudinal relaxation iscaused by transverse noise, via the x- or y-axis, withthe intuition that off-diagonal elements of an interactionHamiltonian are needed to connect and drive transitionsbetween states |0〉 and |1〉.

Depolarization occurs due to energy exchange with anenvironment, generally leading to both an “up transitionrate” Γ1↑ (excitation from |0〉 to |1〉), and a “down tran-sition rate” Γ1↓ (relaxation from |1〉 to |0〉). Together,these form the longitudinal relaxation rate Γ1:

Γ1 ≡1T1

= Γ1↓ + Γ1↑. (44)

T1 is the 1/e decay time in the exponential decay func-tion in Eq. (43), and it is the characteristic time scaleover which qubit population will relax to its steady-state value. For superconducting qubits, this steady-state value is generally the ground state, due to Boltz-mann statistics and typical operating conditions. Boltz-mann equilibrium statistics lead to the “detailed bal-ance” relationship Γ1↑ = exp(−~ωq/kBT )Γ1↓, where Tis the temperature and kB is the Boltzmann constant,with an equilibrium qubit polarization approaching p =tanh(~ωq/2kBT ). Typical qubits are designed at fre-quency ωq/2π ≈ 5 GHz and are operated at dilutionrefrigerator temperatures T ≈ 20 mK. In this limit, theup-rate Γ1↑ is exponentially suppressed by the Boltz-mann factor exp(−~ωq/kBT ), and so only the down-rate Γ1↓ contributes significantly, relaxing the popula-tion to the ground state. Thus, qubits generally spon-taneously lose energy to their cold environment, but the

environment rarely introduces a qubit excitation. As aresult, the equilibrium polarization approaches unity [seeEq. (43)]117,118.

Only noise at the qubit frequency mediates qubit tran-sitions, whether absorption or emission, and this noise isgenerally “well behaved” (short correlation time, manymodes weakly coupled to qubit, no divergences) aroundthe qubit frequency for superconducting qubits. The in-tuition is that qubit-transition linewidths are relativelynarrow in frequency, and so the noise generally does notvary much over this narrow frequency range. Althoughthere are a few notable exceptions, for example, qubitdecay in the presence of hot quasiparticles119–121, whichcan lead to non-exponential decay functions, longitudinaldepolarization measurements generally present exponen-tial decay functions consistent with the Bloch-Redfieldpicture.

An example of a T1 measurement is shown in Fig. 5(a).The qubit is prepared in its excited state using an Xπ-pulse, and then left to spontaneously decay to the groundstate for a time τ , after which the qubit is measured. Asingle measurement will project the quantum state intoeither state |0〉 or state |1〉, with probabilities that cor-respond to the qubit polarization. To make an estimateof this polarization, one needs to identically prepare thequbit and repeat the experiment many times. This isanalogous to flipping a coin: any single flip will yieldheads or tails, but the probability of obtaining a headsor tails can be estimated by flipping the coin many timesand taking the ensemble average. The resulting expo-nential decay has a characteristic time T1 = 85 µs.Pure dephasing

The pure dephasing rate Γφ describes depolarizationin the x − y plane of the Bloch sphere. It is referred toas “pure dephasing,” to distinguish it from other phase-breaking processes such as energy excitation or relax-ation.

As illustrated in Fig. 4(c), pure dephasing is causedby longitudinal noise that couples to the qubit via the z-axis. Such longitudinal noise causes the qubit frequencyωq to fluctuate, such that it is no longer equal to therotating frame frequency ωd, and causes the Bloch vec-tor to precess forward or backward in the rotating frame.Intuitively, we can imagine identically preparing severalinstances of the Bloch vector along the x-axis. For eachinstance, the stochastic fluctuations of qubit frequencywill result in a different precession frequency, resultingin a net fanout of the Bloch vector in the x − y plane.This eventually leads to a complete depolarization of theazimuthal angle φ. Note that this stochastic effect will becaptured in the transverse relaxation rate Γ2 (next sec-tion); it is not the deterministic term exp(±iδωt) thatappears in Eq. (43), which represents intentional detun-ing of the qubit reference frame.

There are a few important distinctions between puredephasing and energy relaxation. First, in contrast toenergy relaxation, pure dephasing is not a resonant phe-nomenon; noise at any frequency can modify the qubit

15

Relaxation:

ReadoutXπ

τ

t Echo:

Xπ/2τ/2

t

Xπ/2Xπ

τ/2

Ramsey:Xπ/2 τ

tXπ/2

(a)

(b)

(c)

(d)

FIG. 5. Characterizing longitudinal (T1) and transverse (T2) relaxation times of a transmon qubit116. (a) Longitudinalrelaxation (energy relaxation) measurement. The qubit is prepared in the excited state using an Xπ-pulse and measured aftera waiting time τ . For each value τ , this procedure is repeated to obtain an ensemble average of the qubit polarization: +1corresponding to |0〉, and −1 corresponding to |1〉. The resulting exponential decay function has a characteristic time T1 = 85µs. (b) Transverse relaxation (decoherence) measurement via Ramsey interferometry. The qubit is prepared on the equatorusing an Xπ/2-pulse, intentionally detuned from the qubit frequency by δω, causing the Bloch vector to precess in the rotatingframe at a rate δω around the z-axis. After a time τ , a second Xπ/2 pulse then projects the Bloch vector back on to thez axis, effectively mapping its former position on the equator to a position on the z axis. The oscillations decay with anapproximately (but not exactly) exponential decay function, with a characteristic time T ∗2 = 95 µs. (c) Transverse relaxation(decoherence) measurement via a Hahn echo experiment115. The qubit is prepared and measured in the same manner as theRamsey interfometry experiment, except that a single Xπ pulse is applied midway through the free-evolution time τ . Thedecay function is approximately exponential, with a characteristic time T2E = 120 µs. The coherence improvement using theHahn echo over panel (b) indicates that some low-frequency dephasing noise has been mitigated; however, a small amountremains since T2E has not yet reached the 2T1 limit. (d) Coherence function incorporating T1 loss and Gaussian dephasingcomponents of the Ramsey interferometry data in panel (b). The Gaussian-distributed 1/f noise spectrum of magnetic fluxnoise leads to a decay function exp(−χN ) ∝ exp(−t/T1) exp(−t2/T 2

ϕ,G) in Eq. (45). These two decay functions together matchwell the Ramsey data in panel (b).

frequency and cause dephasing. Thus, qubit dephasingis subject to broadband noise. Second, since pure de-phasing is elastic (there is no energy exchange with theenvironment), it is in principle reversible. That is, thedephasing can be “undone” – with quantum informa-tion being preserved – through the application of unitaryoperations, e.g., dynamical decoupling pulses78, see Sec.III D 2.

The degree to which the quantum information can beretained depends on many factors, including the band-

width of the noise, the rate of dephasing, the rate atwhich unitary operations can be performed, etc. Thisshould be contrasted with spontaneous energy relaxation,which is an irreversible process. Intuitively, once thequbit emits energy to the environment and its myriaduncontrollable modes, the quantum information is essen-tially lost with no hope for its recovery and reconstitutionback into the qubit.

Transverse relaxationThe transverse relaxation rate Γ2 = Γ1/2 + Γϕ describes

16

the loss of coherence of a superposition state, for example(1/√

2)(|0〉+|1〉), pointed along the x-axis on the equatorof the Bloch sphere as illustrated in Fig. 4(d). Decoher-ence is caused in part by longitudinal noise, which fluctu-ates the qubit frequency and leads to pure dephasing Γϕ(red). It is also caused by transverse noise, which leadsto energy relaxation of the excited-state component ofthe superposition state at a rate Γ1 (blue). Such a relax-ation event is also a phase-breaking process, because onceit occurs, the Bloch vector points to the north pole, |0〉,and there is no longer any knowledge of which directionthe Bloch vector had been pointing along the equator;the relative phase of the superposition state is lost.

Transverse relaxation T2 can be measured using Ram-sey interferometry, as shown and described in Fig. 5(b).The protocol positions the Bloch vector on the equatorusing a Xπ/2-pulse. Typically, the carrier frequency ofthis pulse is slightly detuned from the qubit frequencyby an amount δω. As a result, the Bloch vector will pre-cess around the z-axis at a rate δω. This is done forconvenience sake, so that the resulting Ramsey measure-ment will oscillate, making it easier to analyze. Afterprecessing for a time τ , a second Xπ/2-pulse projects theBloch vector back on to the z-axis. Repeated measure-ments are made to take an ensemble averaged estimate ofthe qubit polarization, as a function of τ . The resultingoscillations in Fig. 5(b) feature an approximately expo-nential decay function with time T ∗2 = 98 µs. The “*”indicates that the Ramsey experiment is sensitive to in-homogeneous broadening. That is, it is highly sensitiveto quasi-static, low-frequency fluctuations that are con-stant within one experimental trial, but vary from trialto trial, e.g., due to 1/f -type noise. This sensitivity toquasi-static noise is related to the corresponding N = 0noise filter function shown in Fig. 5(d) being centeredat at zero-frequency, as described in more detail in Sec-tion III D 2.

The Hahn echo shown in Fig. 5(c) is an experimentthat is less sensitive to quasi-static noise. By placinga Yπ pulse at the center of a Ramsey interferometry ex-periment, the quasi-static contributions to dephasing canbe “refocused,” leaving an estimate T2E that is less sen-sitive to inhomogeneous broadening mechanisms. Thepulses are generally chosen to be resonant with the qubittransition for a Hahn echo, since any frequency detuningwould be nominally refocused anyway. The resulting de-cay function in Fig. 5(c) is essentially exponential withtime T2E = 120 µs.

With the known T1 and T2 times, one can infer the puredephasing time Tϕ from Eq. (41), provided the decayfunctions are exponential. In superconducting qubits,however, the broadband dephasing noise (e.g., flux noise,charge noise, critical-current noise, ...) tends to exhibita 1/f -like power spectrum. Such noise is singular nearω = 0, has long correlation times, and generally doesnot fall within the Bloch-Redfield description. The de-cay function of the off-diagonal terms in Eq. (43) aregenerally non-exponential, and for such cases, the simple

expression in Eq. (41) is not applicable.

3. Modification due to 1/f-type noise

If we assume that the qubit is coupled to many in-dependent fluctuators, then, regardless of their individ-ual statistics, they will in concert generate noise witha Gaussian distribution due to the central limit theo-rem. We therefore say that the longitudinal fluctuationsexhibit Gaussian-distributed 1/f noise122,123. For 1/fnoise spectra, the phase decay function is itself a Gaus-sian exp

[−(t/Tϕ,Gt)2], where we write Tϕ,G to distin-

guish it from Tϕ used in Eq. (41). Furthermore, thisfunction is separable from the T1-type exponential de-cay, because the T1-noise remains regular at the qubitfrequency. The density matrix in Eq. (43) becomes, fol-lowing Refs. 78 and 112,

ρ =(

1 + (|α|2 − 1)e−Γ1t αβ∗eiδωte−Γ12 te−χN (t)

α∗βe−iδωte−Γ12 te−χN (t) |β|2e−Γ1t

),

(45)where the decay function 〈exp(−χN (t))〉 contains the co-herence function χN (t), which generalizes pure dephas-ing to include non-exponential decay functions. As weshall see later, the subscript N labeling the decay func-tion refers to the number of π-pulses used to refocus thelow-frequency noise, which impacts the form of the de-cay function. Because the function is no longer purelyexponential, we cannot formally write the transverse re-laxation decay function as exp(−t/T2). However, an ex-ponential decay remains a practically reasonable approx-imation for Tϕ & T1. We also note that the energy decaycomponent of the transverse relaxation is exp(−t/2T1),and so T2 can never be larger than 2T1. In the absenceof pure dephasing, the maximum T2 = 2T1 is reached.

As an example, consider the Ramsey interferometrydata in Fig. 5(b). Since the dephasing is relatively weak,the transverse relaxation function as exp(−t/T2) is areasonable fit and yields T2 = 95 µs. However, us-ing the value T1 = 85 µs from Fig. 5(a) and dividingout exp(−t/2T1) from the data in Fig. 5(b), the remain-ing pure dephasing decay function is shown in Fig. 5(d)and assumes a Gaussian envelope 〈exp(−χN (t))〉 =exp

[−(t/Tϕ,Gt)2], with Tϕ,G = 98 µs. The Hahn echo

data in Fig. 5(c) may be treated similarly.For completeness, in addition to 1/f dephasing mech-

anisms, we note that there are also “white” pure dephas-ing mechanisms, which give rise to an exponential de-cay function for the dephasing component of T2. Onecommon example is dephasing due to the shot noise ofresidual photons in the readout resonator coupled to su-perconducting qubits, as we discuss in Section III C 3.

17

Qubit absorptionfrom environmentQubit absorptionfrom environment

Qubit emissionto environmentQubit emissionto environment

Thermal (Johnson)

noise Quantum (Nyquist)

noise

“Low-frequency” noise

1/f noise

Johnson-Nyquist

noise

FIG. 6. Examples of symmetric and asymmetric noise spec-tral densities. Noise at positive (negative) frequencies cor-responds to the qubit emitting (absorbing) energy to (from)its environment. Thermal noise is proportional to temper-ature T and carries essentially a white noise spectrum. Asit represents a classical fluctuating parameter, such as elec-tric current, the noise power spectral density is symmetricin frequency. When resonant with the qubit, it will driveboth stimulated emission and absorption processes. The qubitmay also spontaneously emit energy to its environment, rep-resented as Nyquist noise124, a quantum mechanical effectthat is not symmetric in frequency. At sufficiently low tem-peratures or high frequencies, ~ω > 2kBT , the Nyquist noisedominates thermal noise. Another common example is 1/fnoise, which is also a classical noise fluctuation and symmet-ric in frequency.

4. Noise power spectral density (PSD)

The frequency distribution of the noise power for a sta-tionary noise source λ is characterized by its PSD Sλ(ω)

Sλ(ω) =∫ ∞−∞

dτ 〈λ(τ)λ(0)〉e−iωτ . (46)

The Wiener-Khintchine theorem states that the PSDis the Fourier transform of the autocorrelation functioncλ(τ) = 〈λ(τ)λ(0)〉 of the noise source λ. Since the in-tegration limits are (−∞,∞), this is the bilateral PSD.Symmetrizing the PSD allows one to consider only posi-tive frequencies, which is termed a unilateral PSD. Bothunilateral and bilateral PSDs are used, often with thesame notation, and so one needs to know how the PSDis defined, keep track of the factors of 2 and π, and alsobe aware of the implications for quantum systems.

For classical systems, the noise power spectral densityis symmetric. This is because the autocorrelation func-tion of real signals is itself a real function, and the Fouriertransform of a real temporal function is symmetric in thefrequency domain. Dephasing noise is caused by real,fluctuating fields, and so its PSD is generally symmetric.

Examples of such classical noise include thermal (John-son) noise and 1/f noise125 (see Fig. 6).

In turn, the inverse Fourier transform of the PSD willyield the autocorrelation function:

cλ(τ) = 12π

∫ ∞−∞

dω Sλ(ω)eiωτ . (47)

This implies that integrating the noise power spectraldensity with τ = 0 yields the second moment of the noise,or, for zero-mean fluctuations, the variance.

However, the autocorrelation function for a quantumsystem may be complex-valued due to the fact that quan-tum operators generally do not commute at differenttimes. This means that time-ordering of the operatorsmatters, and the PSD need not be symmetric in fre-quency. This is generally the case for transverse noisecausing longitudinal energy relaxation. Noise at a posi-tive frequency S(ωq) corresponds to energy transfer fromthe qubit to the environment, including both stimulatedand spontaneous emission, associated with the down-rateΓ1↓. Noise at a negative frequency S(−ωq) correspondsto energy transfer to the qubit from the environment, as-sociated with the up-rate Γ1↑. For a detailed discussion,see Refs. 126 and 127. Spontaneous emission to a coldenvironment or electromagnetic vacuum, represented byNyquist noise in Fig. 6, is an example of an asymmetricnoise PSD124.

In general, making a connection between Sλ(ω) andthe measured qubit decay functions is the basis for noisespectroscopy up to second-order statistics78,128–131. Thesearch for higher-order spectra related to non-Gaussiannoise is a current topic of active research132.

C. Common examples of noise

There are many sources of stochastic noise in super-conducting qubits, and we refer the reader to Ref. 40 fora review. Here, we briefly present several of the mostcommon types of noise, their affect on coherence, andrefer the reader to the references for a more detailed dis-cussion.

1. Charge noise

Charge noise is ubiquitous in solid-state devices. Itarises from charged fluctuators present in the defects orcharge traps that reside in interfacial dielectrics, the junc-tion tunnel barrier, and in the substrate itself. Theseare often modeled as an ensemble of fluctuating two-levelsystems or as bulk dielectric loss133,134. For example, inthe case of a transmon qubit, the electric field betweenthe capacitor plates traverses and couples to dielectricdefects residing on the metal surfaces of the plates (forlateral-plate-type capacitors) or the capacitor dielectricbetween the plates (for parallel-plate-type capacitors).

18

The electric field variable is transverse with respect tothe quantization axis of the transmon qubit, which meansthat this noise is mainly responsible for energy relaxation(T1). Additionally, if the EJ/EC ratio of the transmon isnot made sufficiently large (smaller than around 60), thequbit frequency itself will also be sensitive to broadbandcharge fluctuations. In this case, low-frequency chargenoise couples longitudinally to the transmon and causespure dephasing (Tϕ).

Charge noise is modeled primarily as a combination ofinverse-frequency noise and Nyquist noise, also referredto as ohmic noise. At lower frequencies, the spectraldensity takes the form

SQ(ω) = A2Q

(2π × 1Hz

ω

)γQ, (48)

with quasi-universal values A2Q = (10−3e)2/Hz at 1 Hz,

and γQ ≈ 1. In addition to large 1/f fluctuations, earlycharge qubits often witnessed discrete, charge offsets rem-iniscent of random telegraph noise. Together, these twomechanisms severely limited the utility of charge qubits,and served as a strong motivation to move to capaci-tively shunted charge qubits (transmons), which greatlyreduced the qubit longitudinal sensitivity to charge noise.At higher frequencies, the power spectrum takes the formSQ(ω) = B2

Q[ω/(2π×1Hz)], where the noise strength B2Q

at 1 Hz can assume a range of values depending on thelevel of dissipation in the system. Likewise, the cross-overfrom 1/f -like behavior to f -like behavior generally occursat around 1 GHz, but will vary higher or lower betweensamples depending on the degree of dissipation62,135.

2. Magnetic flux noise

Another commonly observed noise in solid-state de-vices is magnetic flux noise. The origin of this noise isunderstood to arise from the stochastic flipping of spins(magnetic dipoles) that reside on the surfaces of the su-perconducting metals comprising the qubit136, resultingin random fluctuations of the effective magnetic field thatbiases flux-tunable qubits.

For example, in the case of the split transmon, theexternal magnetic field threading the loop couples longi-tudinally to the qubit and modulates the transition fre-quency via the Josephson energy EJ (except at ϕe = 0,where the qubit is first-order insensitive to magnetic-field fluctuations). Because the flux noise is longitudinalto the transmon, it contributes to pure dephasing (Tϕ).However, in the case of the flux qubit, and depending onthe flux-bias point, the flux noise may be either longitu-dinal – causing dephasing Tϕ – or it may couple trans-versely and thus contribute to T1 relaxation62,78. Thenoise power spectrum of these fluctuations generally ex-hibits a “quasi-universal” dependence,

SΦ(ω) = A2Φ

(2π × 1Hz

ω

)γΦ

, (49)

with γΦ ≈ 0.8 − 1.0 and A2Φ ≈ (1 µΦ0)2/Hz, and has

been shown to extend from less than millihertz to beyondgigahertz frequencies78,130,131,137,138.

The large, low-frequency weighting of the 1/f powerdistribution enables the use of engineered error miti-gation techniques – such as dynamical decoupling – toachieve better coherence78,139–141 and for improving sin-gle and two-qubit gate fidelity142. It was recently demon-strated that 1/f flux noise is also a T1-mechanism whenextended out to the qubit frequency62, and one similarlyexpects a crossover to ohmic flux noise at high enoughfrequencies143.

Although much is known about the statistics andnumber of the defects presumed responsible for fluxnoise, their precise physical manifestation remains uncer-tain136,144. The fact that the 1/f noise is quasi-universaland largely independent of device, strongly suggests acommon origin for the noise. Recent studies suggest thatadsorbed molecular oxygen may be responsible for flux-noise144,145.

3. Photon number fluctuations

In the circuit QED architecture, resonator pho-ton number fluctuation is another major decoherencesource146. Residual microwave fields in the cavity havephoton-number fluctuations that in the dispersive regimeimpact the qubit through an interaction term χσzn, seeSec. II C 2, leading to a frequency shift ∆Stark = 2ηχn,where n is the average photon number, and η = κ2/(κ2 +4χ2) effectively scales the photon population seen by thequbit due to the interplay between the qubit-induced dis-persive shift of the resonator frequency (χ) and the res-onator decay rate (κ).

In the dispersive limit, the noise is longitudinally cou-pled to the qubit and leads to pure dephasing at a rate,

Γφ = η4χ2

κn. (50)

The fluctuations originate from residual photons in theresonator, typically due to radiation from higher tem-perature stages in the dilution refrigerator106,147. Thecorresponding noise spectral density is of a Lorentziantype,

S(ω) = 4χ2 2ηnκω2 + κ2 , (51)

which exhibits an essentially white noise spectrum up toa 3dB cutoff frequency ω = κ set by the resonator decayrate κ, see Ref. 62.

4. Quasiparticles

Quasiparticles, i.e. unpaired electrons, are another im-portant noise source for superconducting devices120. The

19

tunneling of quasiparticles through a qubit junction maylead to both T1 relaxation and pure dephasing Tϕ, de-pending on the type of qubit, the bias point, and thejunction through which the tunneling event occurs119,121.

Quasiparticles are naturally excited due to thermo-dynamics, and the quasiparticle density in equilibriumsuperconductors should be exponentially suppressed astemperature decreases. However, below about 150 mK,the quasiparticle density observed in superconducting de-vices – generally in the range 10−8−10−6 per Cooper pair– is much higher than BCS theory would predict for a su-perconductor in equilibrium with its cryogenic environ-ment at 10 mK. The reason for this excess quasiparticlepopulation is unclear, but it is very likely related to thepresence of additional, non-thermal mechanisms that in-crease the generation rates, “bottleneck effects” that oc-cur at millikelvin temperatures to reduce recombinationrates, or a combination of both.

It has been shown that the observed T1 and excessexcited-state population measured in today’s state-of-the-art high-coherence transmon are self-consistent withexcess “hot” nonequilibrium quasiparticles at the quasi-universal density of around 10−7 − 10−6 per Cooperpair148,149. Although this quasiparticle generation mech-anism is not yet well understood, it has been shown thatquasiparticles can be transiently pumped away, improv-ing T1 times and reducing T1 temporal variation121.

D. Operator form of qubit-environment interaction

Similar to the way that two qubits are coupled, a qubitmay couple and interact with uncontrolled degrees offreedom (DOF) in its environment (the noise sources).The interaction Hamiltonian between the qubit DOF(Oq) and those of the noise source (λ) may be expressedin a general form

Hint = νOqλ (52)

where ν denotes the coupling strength – which is relatedto the sensitivity of the qubit to environmental fluctua-tions ∂Hq/∂λ – and we assume that Oq is a qubit opera-tor within the qubit Hamiltonian Hq. The noisy environ-ment represented by the operator λ produces fluctuationsδλ. Note that we retained the hats in this section to re-mind us that these are quantum operators.

1. Connecting T1 to S(ω)

If the coupling is transverse to the qubit, e.g. Oqis of the type σx or (a + a†) – see the related case ofqubit-qubit coupling treated in Sec. II C – then noiseat the qubit frequency can cause transitions betweenthe qubit eigenstates. Since this is a stochastic process,the ensemble-average manifests itself as a decay (usually

exponential) of the qubit population towards a certainequilibrium value (usually the qubit ground state |0〉 forkBT ~ωq). Again, this process is equivalently referredto as “T1 relaxation”, “energy relaxation”, or “longitudi-nal relaxation”. As stated above, T1 is the characteristictime scale of the decay. Its inverse, Γ1 = 1/T1 is calledthe relaxation rate and depends on the power spectraldensity of the noise S(ω) at the transition frequency ofthe qubit ω = ωq:

Γ1 = 1~2

∣∣∣∣∣〈0|∂Hq

∂λ|1〉∣∣∣∣∣2

Sλ(ωq), (53)

where ∂Hq/∂λ is the qubit transverse susceptibility tofluctuations δλ, such that |δλ|2 is the ensemble averagevalue of the environmental noise sources as seen by thequbit. Eq. (53) is equivalent to Fermi’s Golden Rule,in which the qubit’s transverse susceptibility to noise isdriven by the noise power spectral density. The qubittransverse susceptibility can be used to calculate the pref-actors; for example, for fluctuations δλ = δn, the rele-vant term in the transmon Hamiltonian in Eq. (16) is4EC(n − ng)2, where we allow for an offset charge ng,and the susceptibility is given by 8EC n. We refer thereader to Refs. 150–152 for more details.

2. Connecting Tϕ to S(ω)

If the coupling to the qubit is instead longitudinal,e.g. Hq is of the type σz or a†a, the noise will stochasti-cally modulate the transition frequency of the qubit andthereby introduce a stochastic phase evolution of a qubitsuperposition state. This gradually leads to a loss ofphase information, and it is therefore called pure dephas-ing (time constant Tϕ). Unlike T1 relaxation, which isgenerally an irreversible (incoherent) error, pure dephas-ing Tϕ is in principle reversible (a coherent error). Thedegree of pure dephasing depends on the control pulsesequence applied while the qubit is subject to the noiseprocess.

Consider the relative phase ϕ of a superposition stateundergoing free evolution in the presence of noise. Thesuperposition state’s accumulated phase,

ϕ(t) =∫ t

0ωqdt

′ = 〈ωq〉t+ δϕ(t) (54)

diffuses due to adiabatic fluctuations of the transitionfrequency,

δϕ(t) = ∂ωq

∂λ

∫ t

0δλ(t′)dt′, (55)

where ∂ωq/∂λ = (1/~)|〈∂Hq/∂λ〉| is the qubit’s longi-tudinal sensitivity to λ-noise. For noise generated by alarge number of fluctuators that are weakly coupled to

20

N -pulses (N 1)

2N N N N N N 2N

t

CP / CPMG2 2

0 2 4 6 8Frequency, f (MHz)

0

1

g

(

)

N

sN = 0

12

106

1 10 100 1000Number of pulses, N

T

( s

)

0.1

1

10

CPMG

CPMG simulation

Ramsey (N=0)

2T1

(a)

(b)

>

S~1/f

CPMG

2

FIG. 7. Dynamical error suppression. (a) Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence applies N equallyspaced π pulses within an otherwise free-evolution time τ .Pulses in the time domain correspond to bandpass filters inthe frequency domain (lower panel) which serve to shape thenoise power spectrum seen by the qubit. The centroid of thebandpass filter shifts to higher frequencies as N is increased.For noise that decreases with frequency, such as 1/f noise,larger N corresponds to less integrated noise impinging onthe qubit. (b) CPMG pulse sequence applied to a flux qubitbiased at a point that is highly sensitive to 1/f flux noise. TheRamsey (N = 0) time is approximately 300 ns, and the Hahnecho (N = 1) time is approximately 1.5 µs. Increasing thenumber of CPMG pulses continues to increase the effectiveT2 time towards the 2T1 limit. Adapted from Ref. 78.

the qubit, its statistics are Gaussian. Ensemble averagingover all realizations of the Gaussian-distributed stochas-tic process δλ(t), the dephasing is

〈ei δϕ(t)〉 = e−12 〈δϕ

2(t)〉 ≡ e−χN (t), (56)

leading to a coherence decay function,

〈e−χN (τ)〉 = exp[−τ

2

2∂ωq

∂λ

∫ ∞−∞

gN (ω, τ)S(ω)dω],

(57)where g(ω, τ) is a dimensionless weighting function.

The function gN (ω, τ) can be viewed as a frequency-domain filter of the noise Sλ(ω) [see Fig. 7(a)]. In gen-eral, its filter properties depend on the number N anddistribution of applied pulses. For example, consideringsequences of π-pulses78,153–157,

gN (ω, τ) = 1(ωτ)2

∣∣∣1 + (−1)1+N exp(iωτ)+

2N∑j=1

(−1)j exp(iωδjτ) cos(ωτπ/2)∣∣∣2, (58)

where δj ∈ [0, 1] is the normalized position of the centreof the jth π-pulse between the two π/2-pulses, τ is thetotal free-induction time, and τπ is the length of eachπ-pulse156,157, yielding a total sequence length τ +Nτπ.As the number of pulses increases for fixed τ , the filterfunction’s peak shifts to higher frequencies, leading to areduction in the net integrated noise for 1/fα-type noisespectra with α > 0. Similarly, for a fixed N , the filterfunction will shift in frequency with τ . Additionally, for afixed time separation τ ′ = τ/N (valid for N ≥ 1), the fil-ter sharpens and asymptotically peaks at ω′/2π = 1/2τ ′as more pulses are added. gN (ω, τ) is thus called the “fil-ter function”78,155, and it depends on the pulse sequencesbeing applied. From Eq. (57), the pure dephasing de-cay arises from a noise spectral density that is “shaped”or “filtered” by the sequence-specific filter function. Bychoosing the number of pulses, their rotation axes, andtheir arrangement in time, we can design filter functionsthat minimize the net noise power for a given noise spec-tral density within the experimental constraints of the ex-periment (e.g., pulse-modulation bandwidth of the elec-tronics used to control the qubits).

To give a standard example, we compare the coherenceintegral for two cases: a Ramsey pulse sequence and aHahn echo pulse sequence. Both sequences involve twoπ/2 pulses separated by a time τ , during which free evo-lution of the qubit occurs in the presence of low-frequencydephasing noise. The distinction is that the Hahn echowill place a single π pulse (N = 1) in the middle of thefree-evolution period, whereas the Ramsey does not useany additional pulses (N = 0). The resulting filter func-tions are:

g0(ω, τ) = sinc2ωτ

2 (59)

g1(ω, τ) = sin2 ωτ

4 sinc2ωτ

4 (60)

where the subscript N = 0 and N = 1 indicate the num-ber of π-pulses applied for the Ramsey and Hahn echoexperiments, respectively. The filter function g0(ω, τ) for

21

the Ramsey case is a sinc-function centered at ω = 0. Fornoise that decreases with frequency, e.g., 1/f flux noisein superconducting qubits, the Ramsey experiment win-dows through the noise in S(ω) where it has its highestvalue. This is the worst choice of filter function for 1/fnoise. In contrast, the Hahn echo filter function has acentroid that is peaked at a higher frequency, away fromω = 0. In fact, it has zero value at ω = 0. For noisethat decreases with frequency, such as 1/f noise, this isadvantageous. This concept extends to larger numbersN of π pulses, and is called a Carr-Purcell-Meiboom-Gill(CPMG) sequence158,159. In Fig. 7(b), the T2 time of aqubit under the influence of strong dephasing noise is in-creased toward the 2T1 limit using a CPMG dynamicalerror-suppression pulse sequence with an increasing num-ber of pulses, N . We refer the reader to Refs. 78, 160, and161, where these experiments were performed with super-conducting qubits.

3. Noise spectroscopy

The qubit is highly sensitive to its noisy environment,and this feature can be used to map out the noise powerspectral density. In general, one can map the noise PSDduring free evolution – periods of time for which no con-trol is applied to the qubit, except for very short dynami-cal decoupling pulses – and during driven evolution – pe-riods of time during which the control fields are appliedto the qubit. Both free-evolution and driven-evolutionnoise is important to characterize, as the noise PSD maydiffer for these two types of evolution, and both are uti-lized in the context of universal quantum computation.We refer the reader to Ref. 131 for a summary of noisespectroscopy during both types of evolution.

The Ramsey frequency itself is sensitive to longitudinalnoise, and monitoring its fluctuations is one means tomap out the noise spectral density over the sub-millihertzto ∼ 100 Hz range130,162.

At higher frequencies, the CPMG dynamical decou-pling sequence can be used to create narrow-band fil-ters that “sample” the noise at different frequencies as afunction of the free-evolution time τ and the number ofpulses N . This has been used to map out the noise PSDin the range 0.1 - 300 MHz78. One must be careful ofthe additional small peaks at higher-frequencies, whichall contribute to the dephasing used to perform the noisespectroscopy163.

In fact, using pulse envelopes such as Slepians164 –which are designed to have concentrated frequency re-sponse – to perform noise spectroscopy is one means toreduce such errors156.

At even higher frequencies, measurements of T1 can beused in conjunction with Fermi’s golden rule to map outthe transverse noise spectrum above 1 GHz62,78,165.

The aforementioned are all examples of noise spec-troscopy during free evolution. Noise spectroscopy dur-ing driven evolution was also demonstrated using a “spin-

locking” technique, where a strong drive along x or yaxes defines a new qubit quantization axis, whose Rabifrequency is the new qubit frequency in the spin-lockingframe. The spin-locking frame is then used to infer thenoise spectrum while the qubit is continually subject toa driving field. For more information, we refer the readerto Ref. 131.

E. Engineering noise mitigation

Here, we briefly review a few examples of techniquesthat have been developed to reduce noise or reduce its im-pact on decoherence (sensitivity). We stress that improv-ing gate fidelity is a comprehensive optimization task, onethat is full of trade-offs. It is thus important to identifywhat the limiting factors are, what price we have to payto diminish these limiting factors, and what advantage wecan achieve until reaching a better trade-off. These allrequire an accurate understanding the limitations on thegate fidelity, the sources of decoherence, the propertiesof the noise, and how it affects the system performance.

1. Materials and fabrication improvements

Numerous efforts have been undertaken to reducenoise-induced defects due to materials and fabrica-tion40,166. In the case of charge noise, significant effortshave been made to reduce the number of defects, suchas substrate cleaning59,167, substrate annealing168, andtrenching41,61. In the case of flux noise, several groupshave performed experiments to characterize the behaviorand properties of magnetic-flux defects136,169,170. Morerecently, a number of groups have tried optical surfacetreatments to remove these defects144.

In the context of residual quasiparticles, it has beenshown that adding quasiparticle traps to the circuit de-sign can reduce the quasiparticle number, particularly indevices that create excess quasiparticles, such as classi-cal digital logic or operation in the presence of thermalradiation171

2. Design improvements

Another strategy is to reduce qubit sensitivity to thenoise by design. A qubit can only lose energy to defectsif it couples to them. It has been demonstrated thataltering the capacitor geometry to increase the electric-field mode volume reduces the electric field density in thethin dielectric regions that cause loss. This effectivelyreduces the “participation” of the defects and makes thequbits less senstivie to these noise sources.55,62,133.

In another example, the split transmons built usingasymmetric junctions have lower sensitivity to flux noisethan their symmetric counterparts at the expense of de-creased frequency tunability69. This is a good trade-off

22

to make, because generally one is interested in tuning thequbit frequency over a somewhat restricted range (typ-ically around 1 GHz) about the qubit frequency. Whensuch asymmetric transmons are used in a gate schemesuch as the adiabatic CPHASE-gate65, (see Sec.IV F) thequbit is less sensitive to flux noise, has a lower dephasingrate, and this should improve the gate fidelity in general.

3. Dynamical error suppression

As introduced in the previous section, it is advan-tageous to leverage the 1/ω distribution of flux noise,wherein a considerable amount of the noise power re-sides at low frequencies, and so the noise is “quasi-static”.The spin-echo technique115, which disrupts the free evo-lution by a π-pulse, is extremely effective in mitigatingthe pure dephasing by refocusing the coherent phase dis-persion due to low-frequency noise. The more advancedversions, such as the CPMG-sequence, use multiple π-pulses to interrupt the system more frequently, pushingthe filter band to even higher frequencies – a techniqueknown as dynamical decoupling78.

Returning to excess quasiparticles, it has been shownthat quasiparticles can be stochastically pumped awayfrom the qubit region, resulting in longer, and more sta-ble T1 times121. Although the pumping technique usesa series of π-pulses, this technique differs from dynami-cal error suppression of coherent errors in that pulses arestochastically applied, and that it addresses incoherenterrors (T1).

4. Cryogenic engineering

In the case of photon shot-noise, in addition to ap-plying dynamical decoupling techniques, there have beenseveral recent works aimed at reducing the thermal pho-ton flux that reaches the device. This include optimizingthe attenuation of the cryogenic setup106,148,172, remak-ing the cryogenic attenuators with more efficient heatsinking147, adding absorptive “black” material to absorbstray thermal photons173,174, and adding additional cav-ity filters for thermalization175.

IV. QUBIT CONTROL

In this section, we will introduce how superconduct-ing qubits are manipulated to implement quantum al-gorithms. Since the transmon-like variety of supercon-ducting qubits has so far been the most widely deployedmodality for implementing quantum programs, the dis-cussion throughout this section will be focused on moderntechniques for transmons. Nonetheless, the techniquesintroduced here are applicable to all types of supercon-ducting qubits.

We start with a brief review of the gates used in clas-sical computing as well as quantum computing, and theconcept of universality. Subsequently we discuss the mostcommon technique of driving single qubit gates via acapacitive coupling of a microwave line, coupled to thequbit. We introduce the notion of “virtual Z gates” and“DRAG” pulsing. In the latter part of this section, we re-view some of the most common implementations of two-qubit gates in both tunable and fixed-frequeny transmonqubits. The single-qubit and two-qubit operations to-gether form the basis of many of the medium-scale su-perconducting quantum processors that exist today.

Throughout this section, we write everything in thecomputational basis |0〉, |1〉 where |0〉 is the +1 eigen-state of σz and |1〉 is the −1 eigenstate. We use capi-talized serif-fonts to indicate the rotation operator of aqubit state, e.g. rotations around the x-axis by an angleθ is written as

Xθ = RX(θ) = e−iθ2σx = cos(θ/2)1− i sin(θ/2)σx (61)

and we use the shorthand notation ‘X’ for a full π rotationabout the x axis (and similarly for Y := Yπ and Z := Zπ).

A. Boolean logic gates used in classical computers

Universal boolean logic can be implemented on classi-cal computers using a small set of single-bit and two-bitgates. Several common classical logic gates are shown inFig. 8 along with their truth tables. In classical booleanlogic, bits can take on one of two values: state 0 or state1. The state 0 represents logical FALSE, and state 1 rep-resents logical TRUE.

Beyond the trivial “identity operation,” which simplypasses a boolean bit unchanged, the only other possiblesingle-bit boolean logic gate is the NOT gate. As shownin Fig. 8, the NOT gate flips the bit: 0 → 1 and 1 → 0.This gate is reversible, because it is trivial to determinethe input bit value given the output bit values. As wewill see, for two-bit gates, this is not the case.

There are several two-bit gates shown in Fig. 8. Atwo-bit gate takes two bits as inputs, and it passes asan output the result of a boolean operation. One com-mon example is the AND gate, for which the output is1 if and only if both inputs are 1; otherwise, the outputis 0. The AND gate, and the other two-bit gates shownin Fig. 8, are all examples of irreversible gates; that is,the input bit values cannot be inferred from the outputvalues. For example, for the AND gate, an output of log-ical 1 uniquely identifies the input 11, but an output of 0could be associated with 00, 01, or 10. Once the opera-tion is performed, in general, it cannot be “undone” andthe input information is lost. There are several variantsof two-bit gates, including,

• AND and OR;

• NAND (a combination of NOT and AND) and NOR(a combination of NOT and OR);

23

The output is 1 when the input is 0 and 0 when the input is 1.

Input

0

1

Output

1

0

Input

0

0

1

1

0

1

0

1

Output

0

0

0

1

NOT

The output is 1 only when both inputs are 1, otherwise the output is 0.

AND

GATECIRCUITSYMBOL TRUTH TABLE

The output is 0 only when both inputs are 0, otherwise the output is 1.

OR

The output is 0 only when both inputs are 1, otherwise the output is 1.

NAND

The output is 1 only when both inputs are 0, otherwise the output is 0.

NOR

The output is 1 only when the two inputs have di!erent value, otherwise the output is 0.

XOR

The output is 1 only when the two inputs have the same value, otherwise the output is 0.

XNOR

Input

0

0

1

1

0

1

0

1

Output

0

1

1

1

Input

0

0

1

1

0

1

0

1

Output

1

1

1

0

Input

0

0

1

1

0

1

0

1

Output

1

0

0

0

Input

0

0

1

1

0

1

0

1

Output

0

1

1

0

Input

0

0

1

1

0

1

0

1

Output

1

0

0

1

FIG. 8. Classical single-bit and two-bit boolean logic gates.For each gate, the name, a short description, circuit repre-sentation, and input/output truth tables are presented. Thenumerical values in the truth table correspond to the classicalbit values 0 and 1. Adapted from Ref. 176.

• XOR (exclusive OR) and NXOR (NOT XOR).

The XOR gate is interesting, because it is a parity gate.That is, it returns a logical 0 if the two inputs are the

same values (i.e., they have the same parity), and it re-turns a logical 1 if the two inputs have different values(i.e., different parity). Still, the XOR and NXOR gatesare not reversible, because knowledge of the output doesnot allow one to uniquely identify the input bit values.

The concept of universality refers to the ability to per-form any boolean logic algorithm using a small set ofsingle-bit and two-bit gates. A universal gate set can inprinciple transform any state to any other state in thestate space represented by the classical bits. The set ofgates which enable universal computation is not unique,and may be represented by a small set of gates. For ex-ample, the NOT gate and the AND gate together forma universal gate set. Similarly, the NAND gate itself isuniversal, as is the NOR gate. The efficiency with whichone can implement arbitrary boolean logic, of course, de-pends on the choice of the gate set.

B. Quantum logic gates used in quantum computers

Quantum logic can similarly be performed by a smallset of single-qubit and two-qubit gates. Qubits can ofcourse assume the classical states |0〉 and |1〉, at the northpole and south pole of the Bloch sphere, but they can alsoassume arbitrary superpositions α|0〉+β|1〉, correspond-ing to any other position on the sphere.

Single-qubit operations translate an arbitrary quan-tum state from one point on the Bloch sphere to anotherpoint by rotating the Bloch vector (spin) a certain an-gle about a particular axis. As shown in Fig. 9, thereare several single-qubit operations, each represented bya matrix that describes the quantum operation in thecomputational basis represented by the eigenvectors ofthe σz operator, i.e. |0〉 ≡ [1 0]T and |1〉 ≡ [0 1]T .

For example, the identity gate performs no rotation onthe state of the qubit. This is represented by a two-by-two identity matrix. The X-gate performs a π rotationabout the x axis. Similarly, the Y-gate and Z-gate per-form a π rotation about the y axis and z axis, respec-tively. The S-gate performs a π/2 rotation about the zaxis, and the T-gate performs a rotation of π/4 aboutthe z axis. The Hadamard gate H is also a commonsingle-qubit gate the performs a π rotation about an axisdiagonal in the x-z plane, see Fig. 9.

Two-qubit quantum-logic gates are generally condi-tional gates and take two qubits as inputs. Typically,the first qubit is the control qubit, and the second is thetarget qubit. A unitary operator is applied to the targetqubit, dependent on the state of the control qubit. Thetwo common examples shown in Fig. 10 are the controlledNOT (CNOT-gate) and controlled phase (CZ or CPHASEgate). The CNOT-gate flips the state of the target qubitconditioned on the control qubit being in state |1〉. TheCPHASE-gate applies a Z gate to the target qubit, condi-tioned on the control qubit being in state |1〉. As we willshown later, the iSWAP gate – another two-qubit gate –can be built from the CNOT-gate and single-qubit gates.

24

( )

( )

( )

( )

Identity-gate: no rotation is performed.

Input

|0〉

|1〉

Output

|0〉

|1〉

Input

|0〉

|1〉

Output

|1〉

|0〉

Input

|0〉

|1〉

Output

i |1〉

–i |0〉

Input

|0〉

|1〉

Output

|0〉

–|1〉

Input

|0〉

|1〉

Output

|0〉

|1〉

I I =

gate: rotates the qubit state by π radians (180º) about the x-axis.

X

GATECIRCUIT

REPRESENTATIONMATRIX

REPRESENTATIONTRUTHTABLE

BLOCHSPHERE

I

X

gate: rotates the qubit state by π radians (180º) about the y-axis.

Y

Y

gate: rotates the qubit state by π radians (180º) about the z-axis.

Z

Z

gate: rotates the qubit state by radians (90º) about the z-axis.

S

S

gate: rotates the qubit state by radians (45º) about the z-axis.

T

T

gate: rotates the qubit state by π radians (180º) about an axis diagonal in the x-z plane. This is equivalent to an X-gate followed by a rotation about the y-axis.

H

H

10

01

X = 01

10

Z = 10

0–1

( )H = 11

1–1

S = 10

0

Y = 0i

–i0

( )T = 10

0

Input

|0〉

|1〉

Output

|0〉 + |1〉

Input

|0〉

|1〉

Output

|0〉

|1〉

1

2

ieπ2π

2

π2

π4

ieπ2

ieπ4

2

|0〉 – |1〉

2

ieπ4

z

y

x

z

y

x

z

y

x

z

y

x

z

y

x

90º

45º

( )

z

y

x

180º

180º

z

y

x

180º

180º

FIG. 9. Quantum single-qubit gates. For each gate, the name, a short description, circuit representation, matrix representation,input/output truth tables, and Bloch sphere represenation are presented. Matrices are defined in the basis spanned by thestate vectors |0〉 ≡ [1 0]T and |1〉 ≡ [0 1]T . The numerical values in the truth table correspond to the quantum states |0〉 and|1〉. Adapted from Ref. 176.

The unitary operator of the CNOT gate can be written in a useful way, highlighting that it applies an X depend-

25

Controlled-NOT gate: apply an X-gate to the target qubit if the control qubit is in state |1〉

Input

|00〉

|01〉

|10〉

|11〉

Output

|00〉

|01〉

|11〉

|10〉

Input

|00〉

|01〉

|10〉

|11〉

Output

|00〉

|01〉

|10〉

|11〉

Controlled-phase gate: apply a Z-gate to the target qubit if the control qubit is in state |1〉

GATECIRCUIT

REPRESENTATION

MATRIX

REPRESENTATION

TRUTH

TABLE

Z

CNOT =

1000

0100

0001

0010

CPHASE =

1000

0100

0010

000

–1

FIG. 10. Quantum two-qubit gates: the controlled NOT (CNOT) gate and the controlled phase (CPHASE or CZ). For eachgate, the name, a short description, circuit representation, matrix representation, and input/output truth tables are presented.Matrices are defined in the basis spanned by the two-qubit state vectors |00〉 ≡ [1 0 0 0]T , |01〉 ≡ [0 1 0 0]T , |10〉 ≡ [0 0 1 0]T ,and |11〉 ≡ [0 0 0 1]T , where the first qubit is the control qubit, and the second qubit is the target qubit. The CNOT gate flipsthe state of the target qubit conditioned on the control qubit being in state |1〉. The CPHASE gate applies a Z gate to thetarget qubit conditioned on the control qubit being in state |1〉. Adapted from Ref. 176.

ing on the state of the control qubit.

UCNOT =

1 0 0 00 1 0 00 0 0 10 0 1 0

= |0〉〈0| ⊗ 1+ |1〉〈1| ⊗ X (62)

and similarly for the CPHASE gate,

UCPHASE =

1 0 0 00 1 0 00 0 1 00 0 0 −1

= |0〉〈0| ⊗ 1+ |1〉〈1| ⊗ Z (63)

Comparing the last equality above with the unitary forthe CNOT [Eq. (62)], it is clear that the two gates areclosely related. Indeed, a CNOT can be generated froma CPHASE by applying two Hadamard gates,

UCNOT = (1⊗ H)UCPHASE(1⊗ H), (64)

since HZH = X. Due to the form of Eq. (63), the CPHASEgate is also denoted the CZ gate, since it applies a con-trolled Z operator, by analogy with CNOT (a controlledapplication of X operator). Inspection of the definition ofCPHASE in Fig. 10 makes no distinction between whichqubit acts as the target and which as the control and,consequently, the circuit-diagram is sometimes drawn ina symmetric fashion

CPHASE = •• (65)

The CNOT in terms of CPHASE can then be realized as

CNOT = •H • H

(66)

Some two-qubit gates such as CNOT and CPHASE arealso called entangling gates, because they can take prod-uct states as inputs and output entangled states. Theyare thus an indispensable component of a universal gateset for quantum logic. For example, consider two qubitsA and B in the following state:

|ψ〉 = 1√2

(|0〉+ |1〉)A |0〉B . (67)

If we perform a CNOT gate, UCNOT, on this state, withqubit A the control qubit, and qubit B the target qubit,the resulting state is (see the truth table in Fig. 10):

UCNOT|ψ〉 = 1√2

(|0〉A|0〉B + |1〉A|1〉B) 6= (. . .)A(. . .)B ,

(68)which is a state that cannot be factored into an isolatedqubit-A component and a qubit-B component. This isone of the two-qubit entangled Bell states, a manifestlyquantum mechanical state.

A universal set of single-qubit and two-qubit gates issufficient to implement arbitrary quantum logic. Thismeans that this gate set can in principle reach any statein the multi-qubit state-space. How efficiently this isdone depends on the choice of quantum gates that com-prise the gate set. We also note that each of the single-qubit and two-qubit gates is reversible, that is, given theoutput state, one can uniquely determine the input state.As we discuss further, this distinction between classicaland quantum gates arises, because quantum gates arebased on unitary operations U . If a unitary operation Uis a particular gate applied to a qubit, then its hermitianconjugate U† can be applied to recover the original state,since U†U = I resolves an identity operation.

26

Classical NOT gate

in out

b

b

b

Quantum X gate

b

(a) (b)

FIG. 11. Comparison of the classical inverter (NOT) gateand quantum bit flip (X) gate. (a) The classical NOT gatethat inverts the state of a classical bit. (b) The quantum Xgate, which flips the amplitudes of the two components of aquantum bit.

C. Comparing classical and quantum gates

The gate-sequences used to represent quantum algo-rithms have certain similarities to those used in classi-cal computing, with a few striking differences. As anexample, we consider first the classical NOT gate (dis-cussed previously), and the related quantum circuit ver-sion, shown in Fig. 11.

While the classic bit-flip gate inverts any input state,the quantum bit-flip does not in general produce the an-tipodal state (when viewed on the Bloch sphere), butrather exchange the prefactors of the wavefunction writ-ten in the computational basis. The X operator is some-times referred to as ‘the quantum NOT’ (or ‘quantumbit-flip‘), but we note that X only acts similar to theclassical NOT gate in the case of classical data stored inthe quantum bit, i.e. X|g〉 = |g〉 for g ∈ 0, 1.

As briefly mentioned in Sec. IV B, all quantum gatesare reversible, due to the underlying unitary nature ofthe operators implementing the logical operations. Cer-tain other processes used in quantum information pro-cessing, however, are irreversible. Namely, measurements(see Sec. V for detailed discussion) and energy loss to thethe environment (if the resulting state of the environmentis not known). Here, we will not consider how these pro-cesses are modeled, but refer the interested reader to e.g.Ref. 177, and will only consider unitary control oper-ations throughout the rest of this section. Finally, wenote that quantum circuits are written left-to-right (inorder of application), while the calculation of the resultof a gate-sequences, e.g the circuit

|ψin〉 U0 U1 · · · Un |ψout〉 (69)

is performed right-to-left, i.e.

|ψout〉 = Un · · ·U1U0|ψin〉. (70)

As discussed in Sec. IV A, the NOR and NAND gates areeach individually universal gates for classical computing.Since both of these gates have no direct quantum ana-logue (because they are not reversible), it is natural toask which gates are needed to build a universal quan-tum computer. It turns out that the ability to rotate

about arbitrary axes on the Bloch-sphere (i.e. a completesingle-qubit gate set), supplemented with any entangling2-qubit operation will suffice for universality177,178. Byusing what is known as the ‘Krauss-Cirac decomposition’,any two-qubit gate can be decomposed into a series ofCNOT operations177,179.

1. Gate sets and gate synthesis

A common universal quantum gate set is

G0 = Xθ,Yθ,Zθ,Phθ,CNOT (71)

where Phθ = eiθ1 applies an overall phase θ to a singlequbit. For completeness we mention another universalgate set which is of particular interest from a theoreticalperspective, namely

G1 = H,S,T,CNOT, (72)

As a technical aside, we mention that the restriction toa discrete gate set still gives rise to universality. Thisfact relies on using the so-called Solovay-Kitaev180,181

theorem, which (roughly) states that any other single-qubit gate can be approximated to an error ε using onlyO(logc(1/ε)) (where c > 0) single-qubit gates from G1.The gate-set G1 is typically referred to as the ‘Clifford +T ’ set, where H, S and CNOT are all Clifford gates.

Each quantum computing architecture will have cer-tain gates that are simpler to implement at the hard-ware level than others (sometimes referred to as ’na-tive’ gates of the architecture). These are typically thegates for which the Hamiltonian governing the gate-implementation gives rise to a unitary propagator thatcorresponds to the gate itself. We will show several exam-ples of this in Sections IV E, IV F, and IV G. Regardlessof which gates are natively available, as long as one has acomplete gate set, one can use the Solovay-Kitaev theo-rem to synthesize any other set efficiently. In general onewants to keep the overall number of time steps in whichgates are applied (denoted the depth of a circuit) as lowas possible, and one wants to use as many of the nativegates as possible, to reduce the amount of time spent syn-thesising. Moreover, running a quantum algorithm alsodepends on the qubit connectivity of the device. Theprocess of designing a quantum gate sequence that effi-ciently implements a specific algorithm, while taking intoaccount the considerations outlined above is known asgate synthesis and gate compilation, respectively. A fulldiscussion of this large research effort is outside the scopeof this review, but the interested reader may consult e.g.Refs. 182–184 and references therein as a starting point.As a concrete (and trivial) example of how gate identitiescan be used, in Eq. (73) we illustrate how the Hadamardgate from G1 can be generated by two single-qubit gates(from G0) and an overall phase gate,

27

room temperature on-chipwiring

Vd(t) Rw Cd

C

FIG. 12. Circuit diagram of capacitive coupling of a mi-crowave drive line (characterized by a time-dependent voltageVd(t)) to a generic transmon-like superconducting qubit.

H = Phπ2Yπ

2Zπ = i

1√2

[1 −11 1

] [−i 00 i

]= 1√

2

[1 11 −1

](73)

As we show in Sec. IV D 1, the gates Xθ, Yθ and Zθare all natively available in a superconducting quantumprocessor.

We now address the question of how single qubit ro-tations and two-qubit operations are implemented intransmon-based superconducting quantum processors.

2. Addressing superconducting qubits

The modes of addressing transmon-like superconduct-ing qubits can roughly be split into two main categories:i) Capacitive coupling between a resonator (or a feedline)and the superconducting qubit dipole-field allows for mi-crowave control to implement single-qubit rotations (seeSec. IV D) as well as certain two-qubit gates (see Sec-tions IV G and IV G 4). ii) For flux-tunable qubits, localmagnetic fields can be used to tune the frequency of indi-vidual qubits. This allows the implementation of z-axissingle-qubit rotation as well as multiple two-qubit gates(see Sections IV E, IV F and IV H).

D. Single-qubit gates

In this section we will review the steps necessary todemonstrate that capacitive coupling of microwaves to asuperconducting circuit can be used to drive single-qubitgates. To this end we consider coupling a superconduct-ing qubit to a microwave source (sometimes referred toas a ‘qubit drive’) as shown in Fig. 12(a). A full circuitanalysis of the circuit in Fig. 12(a) is beyond the scopeof this review, so here we settle for highlighting the stepsthat elucidate the physics of the qubit/drive coupling.The interested reader may consult a number of lecturesnotes and pertinent theses (e.g. Refs. 44, 162, 185–187).Here we follow Ref. 162.

1. Capacitive coupling for X,Y control

We start by modeling the qubit as an harmonic os-cillator, for which the (classical) circuit Hamiltonian canbe calculated by circuit quantization techniques, startingfrom Kirchoffs laws, and is given by162

H = Q(t)2

2CΣ+ Φ2

2L + Cd

CΣVd(t)Q, (74)

where CΣ = C + Cd is the total capacitance to groundand Q = CΣΦ − CdVd(t) is a renormalized charge vari-able for the circuit. We can now promote the flux andcharge variables to quantum operators and assume weakcoupling to the drive-line, so that Q ≈ Q, and arrive at

H = HLC + Cd

CΣVd(t)Q, (75)

where HLC = Q2/(2C) + Φ2/(2L) and we have keptonly terms that couple to the dynamic variables. Similarto the momentum operator for a harmonic oscillator in(x, p)–space, we can express the charge variable in termsof raising and lowering operators, as done in Sec. II

Q = −iQzpf(a− a†

)(76)

where Qzpf =√~/2Z is the zero-point charge fluctations

and Z =√L/C is the impedance of the circuit to ground.

Thus, the LC oscillator capacitively coupled to a driveline can be written as,

H = ω

(a†a+ 1

2

)− Cd

CΣVd(t)iQzpf

(a− a†

). (77)

Finally, by truncating to the lowest transition of the oscil-lator we can make the replacement a→ σ− and a† → σ+

throughout and arrive at

H = −ωq

2 σz︸ ︷︷ ︸H0

+ ΩVd(t)σy︸ ︷︷ ︸Hd

(78)

where Ω = (Cd/CΣ)Qzpf and ωq = (E1 − E0)/~.§To elucidate the role of the drive, we move into a frame

rotating with the qubit at frequency ωq (also denoted‘the rotating frame’ or the ‘the interaction frame’). Tosee the usefulness of this rotating frame, consider a state|ψ0〉 = (1 1)T /

√2. By the time-dependent Schrodinger

equation this state evolves according to

|ψ0(t)〉 = UH0 |ψ0〉 = 1√2

(eiωqt/2

e−iωqt/2

), (79)

§Starting from a generic qubit Hamiltonian, H0 = E0|0〉〈0| +E1|1〉〈1|, we can rewrite in terms of Pauli matrices, and getH0 = ((E0 + E1)/2)1 − ((E1 − E0)/2)σz . In the main text wehave ignored the constant offset term.

28

where UH0 is the propagator corresponding to H0. Bycalculating e.g. 〈ψ0|σx|ψ0〉 = cos(ωqt) it is evidentthat the phase is winding with a frequency of ωq dueto the σz term. By going into a frame rotating withthe qubit at frequency ωq, the action of the drive canbe more clearly appreciated. To this end we defineUrf = eiH0t = U†H0

and the new state in the rotatingframe is |ψrf(t)〉 = Urf|ψ0〉. The time-evolution in thisnew frame is again found from the Schrodinger equation(using the shorthand ∂t = ∂/∂t),

i∂t|ψrf(t)〉 = i(∂tUrf)|ψ0〉+ iUrf (∂t|ψ0〉) (80)= iUrfU

†rf|ψrf〉+ UrfH0|ψ0〉 (81)

=(iUrfU

†rf + UrfH0U

†rf

)︸ ︷︷ ︸

H0

|ψrf〉 (82)

We can think of the term H0 in the parentheses inEq. (82) as the form of H0 in the rotating frame. Sim-ple insertion shows that H0 = 0 as expected (the ro-tating frame should take care of the time-dependence).However, one could also think of the term in bracketsin Eq. (82) as a prescription for calculating the form ofany Hamiltonian in the rotating frame given by Urf, byreplacing H0 with some other H. In general, we will notfind H = 0.

Returning to Eq. (78), the form of Hd in the rotatingframe is found to be

Hd = ΩVd(t) (cos(ωqt)σy − sin(ωqt)σx) . (83)

We can in general assume that the time-dependent partof the voltage (Vd(t) = V0v(t)) has the generic form

v(t) = s(t) sin(ωdt+ φ) (84)= s(t) (cos(φ) sin(ωdt) + sin(φ) cos(ωdt)) , (85)

where s(t) is a dimensionless envelope function, so thatthe amplitude of the drive is set by V0s(t). Adopting thedefinitions

I = cos(φ) (the ‘in-phase’ component) (86)Q = sin(φ) (the ‘out-of-phase’ component) (87)

the driving Hamiltonian in the rotating frame takes theform

Hd = ΩV0s(t) (I sin(ωdt)−Q cos(ωdt))× (cos(ωqt)σy − sin(ωqt)σx) (88)

Performing the multiplication and dropping fast rotatingterms that will average to zero (i.e. terms with ωq +ωd),known as the rotating wave approximation (RWA), weare left with

Hd = 12ΩV0s(t)

[(−I cos(δωt) +Q sin(δωt))σx

+ (I sin(δωt)−Q cos(δωt))σy]

(89)

where δω = ωq − ωd. Finally, by re-using the definitionsfrom Eq. (85), the driving Hamiltonian in the rotatingframe using the RWA can be written as

Hd = −Ω2 V0s(t)

(0 ei(δωt+φ)

e−i(δωt+φ) 0

). (90)

Equation (90) is a powerful tool for understanding single-qubit gates in superconducting qubits. As a concreteexample, assume that we apply a pulse at the qubit fre-quency, so that δω = 0, then

Hd = −Ω2 V0s(t) (Iσx +Qσy) , (91)

showing that an in-phase pulse (φ = 0, i.e. the I-component) corresponds to rotations around the x-axis,while an out-of-phase pulse (φ = π/2, i.e. the Q-component), corresponds to rotations about the y-axis.As a concrete example of an in-phase pulse, writing outthe unitary operator yields

Uφ=0rf,d (t) = exp

([i

2ΩV0

∫ t

0s(t′)dt′

]σx

), (92)

which depends only on the macroscopic design parame-ters of the circuit as well as the envelope of the basebandpulse s(t) and amplitude V0, which can both be controlledusing arbitrary waveform generators (AWGs). Equation(92) is known as Rabi driving and can serve as a use-ful tool for engineering the circuit parameters needed forefficient gate operation (subject to the available outputvoltage V0). To see this we define the shorthand

Θ(t) = −ΩV0

∫ t

0s(t′)dt′ (93)

which is the angle by which a state is rotated given thecapacitive couplings, the impedance of the circuit, themagnitude V0, and the waveform envelope, s(t). Thismeans that to implement a π-pulse on the x-axis onewould solve the equation Θ(t) = π and output the sig-nal in-phase with the qubit drive. In this language, asequence of pulses (see Fig. 13(a)) Θk,Θk−1, ...Θ0 is con-verted to a sequence of gates operating on a qubit as

Uk · · ·U1U0 = Tk∏

n=0e[− i

2 Θn(t)(Inσx+Qnσy)], (94)

where T is an operator that ensures the pulses are gen-erated in the time-ordered sequence corresponding toUk · · ·U1U0.

In Fig. 13 we outline the typical IQ modulation setupused to generate the pulses used in Eq. (94). Fig. 13(a)shows how a pulse at frequency ωd is generated using alow phase-noise microwave generator (typically denoted‘the local oscillator (LO)’), while the pulse is shaped bycombining the LO signal in an IQ mixer with pulses gen-erated in an AWG. To allow for frequency multiplexing,

29

(a) (b)

to qubits

I Q

y(Q)x(I)

-x(-I)

II QQ

LO

carrier basebandpulses

RFtime

Ampl

itude

-y(-Q)

AWGLO

ωLO ωAWG

Vd(t)

s(t)

s(t)

ωd=ωLO+ωAWG

(c)

FIG. 13. (a) Schematic of a typical qubit drive setup. A mi-crowave source supplies a high-frequency signal (ωLO), whilean arbitrary waveform generator (AWG) supplies a pulse-envelope (s(t)), sometimes with a low frequency component,ωAWG, generated by the AWG. The IQ-mixer combines the twosignals to generate a shaped waveform Vd(t) with a frequencyωd = ωLO ± ωAWG, typically resonant with the qubit. (b) Ex-ample of how a gate sequence is translated into a waveformgenerated by the AWG. Colors indicate I and Q components.(c) The action of a Xπ/2 pulse on a |0〉 state to produce the|−i〉 = 1√

2 (|0〉 − i|1〉) state.

the AWG signal will typically be generated with a low-frequency component, ωAWG, and the LO signal will beoffset, so that ωLO+ωAWG = ωd. By mixing in more thanone frequency ωAWG1, ωAWG2, ... it is possible to addressmultiple qubits (or readout resonators) simultaneously,via the superposition of individual drives.

The I (Q) input of the IQ mixer will multiply the base-band signal to the in-phase (out-of-phase) component ofthe LO. In Fig. 13(b) we schematically show the com-parison between XY gates in a quantum circuit and thecorresponding waveforms generated in the AWG (omit-ting for clarity the frequency ωAWG component). Theinset in Fig. 13(b) shows an example of a gate on theBloch sphere, with indication of (I,Q) axes. More so-phisticated and compact approaches exist to reduce thehardware needed for XY qubit control, relative to thesetup shown in Fig. 13, see e.g.188–190.

2. Virtual Z gate

As we saw in Sec. IV D, the distinction between x–and y–rotations was merely a choice of phase on the mi-crowave signals, and the angle to be rotated is given byΘ(t), both of which are generated using an AWG. Sincethe choice of phase φ has an arbitrary starting point, wecould consider φ → φ + π/2. This would lead to I → Qand Q → −I. Therefore, changing the phase effectivelychanges rotations around x to rotations around y (and

vice-versa, with a change of sign). This is reminiscent ofthe result of applying a Zπ rotation to x– and y–rotations,where ZπXπ = iYπ and ZπYπ = −iXπ. This analogy be-tween shifting a phase of an AWG-generated signal andapplying Z rotations can be utilized to implement virtualZ gates191. As shown by McKay et al., this intuitioncan be formalized via the following example: considerthe case of applying a pulse with an angle θ on the Ichannel (i.e. a Xθ) followed by another θ pulse on the Ichannel, but with a phase φ0 relative to the first pulse(denoted X(φ0)

θ , where X indicates we still use the I chan-nel, but the rotation axis is now an angle φ0 away fromthe x-axis). Using Eq. (94) this corresponds to a pulsesequence

X(φ0)θ Xθ = e−i

θ2 (cos(φ0)σx+sin(φ0)σy)Xθ (95)

= Z−φ0XθZφ0Xθ (96)

from which we see that the effect of the offset phase φ0is to apply Zφ0 . The equality above can be verified witha little trigonometric footwork. The final Z−φ0 is due tothe rotation being in the frame of reference of the qubit.However, since readout is along z-axis (see Sec. V), afinal phase rotation about z will not change the mea-surement outcome. Thus, if one wants to to implementthe gate sequence

· · · Ui Zθ0 Ui+1 Zθ1 Ui+2 · · · (97)

where Ui’s are arbitrary gates, this can be done by re-vising the gate sequence (in the control software for theAWG) and changing the phase of subsequent pulses

· · · Ui U(θ0)i+1 U

(θ0+θ1)i+2 · · · (98)

which reduces the number of overall gates. Moreover,the virtual-Z gates are “perfect”, in the sense that noadditional pulses are required, and the gate takes “zerotime”, and thus the gate fidelity is nominally unity. Aswe show in Sections IV E and IV F, operation of two-qubit gates can incur additional single-qubit phases. Us-ing the virtual-Z strategy, these phases can be cancelledout, leaving a pure two-qubit interaction.

Finally we mention one more salient feature of thevirtual-Z gates. As shown in Ref.63, any single-qubitoperation (up to a global phase) can be written as

U(θ, φ, λ) = Zφ−π2 Xπ2Zπ−θXπ

2Zλ−π2 , (99)

for appropriate choice of angles θ, φ, λ. This means thataccess to a single physical Xπ

2combined with the virtual-

Z gives access to a complete single qubit gate set! An ex-plicit example of Eq. (99) in action is the Hadamard gate,which can be written as H = Zπ

2Xπ

2Zπ

2, but since the Z’s

can be virtual, it is possible to implement Hadamardsas an effective single pulse operation in superconductingqubits.

30

3. The DRAG scheme

In going from Eq. (77) to Eq. (78) we assumed wecould ignore the higher levels of the qubit. However,for weakly anharmonic qubits, such as the transmon (seeSec. II), this may not be a justified assumption, sinceω1→2

q only differs from ωq(≡ ω0→1q ) by the anharmonicity,

α = ω1→2q − ωq, which is negative and typically around

200 to 300 MHz. This situation is sketched in Fig. 14(a-c), where we illustrate how Gaussian pulses with stan-dard deviations σ = 1, 2, 5 ns have spectral contentthat leads to non-zero overlaps with the ω1→2

q = ωq−|α|frequency. This leads to two deleterious effects: (1 ) leak-age errors which take the qubit out of the computationalsubspace, and (2 ) phase errors. Effect 1 can occur be-cause a qubit in the state |1〉 may be excited to |2〉 as a πpulse is applied, or be excited directly from the |0〉, sincethe qubit spends some amount of time in the |1〉 stateduring the π pulse. Effect 2 occurs because the presenceof the drive results in a repulsion between the |1〉 and |2〉levels, in turn changing ω0→1

q as the pulse is applied. Thisleads to the accumulation of a relative phase between|0〉 and |1〉193. The so-called DRAG procedure194–196

(Derivative Reduction by Adiabatic Gate) seeks to com-bat these two effects by applying an extra signal in theout-of-phase component. The trick is to modify the wave-form envelope s(t) according to

s(t)→ s′(t) =

s(t) on I

λs(t)α

on Q, (100)

where λ is a dimensionless scaling parameter, and λ =0 correspond to no DRAG pulse and s(t) is the timederivative of s(t). The theoretically optimal choice forreducing dephasing error is λ = 0.5 and an optimal choicefor reducing leakage error is λ = 1195,197. InterchangingI and Q in Eq. (100) corresponds to DRAG pulsing forthe Q component.

In practice there can be a deviation from these twooptimal values, often due to pulse distortions in the linesleading to the qubits. Typically, randomized benchmark-ing experiments combined with single-shot measurements(see Sec. V) of the |2〉 state is used to determine the op-timal value of λ. The λ = 0.5, 1 tradeoff was demon-strated explicitly in191,198. However, by extending theoriginal DRAG pulse implementation199,200, it is is possi-ble to reduce both errors simultaneously. By introducinga frequency detuning parameter δf to the waveform195

(defined such that δf = 0 corresponds to qubit fre-quency), i.e.

s′δf (t) = s′(t)ei2πδft, (101)

and choosing λ to minimize leakage errors, then phaseerrors can be reduced simultaneously198. Similarly,by a judicious use of the virtual-Z gate, it is alsopossible to reduce phase errors in combination withDRAG pulsing to reduce leakage191. Modern single-qubit

(a)

E0ωq01

E1

2E1α

E2

Ampl

. (a.

u.)

-0.5 -0.25 0.25 0.50Frequency (GHz)

time (ns)

time (ns)Am

pl. (

a.u.

)

-20 -10 0 10Time (ns)

20

Ampl

. (a.

u.)

Ampl

. (a.

u.)

(b)

(c)

(d) (e)

(f) (g)

σ (ns)

FFT

α

λ = 0

λ = 0.5

1

IQ

IQ

25

time

time

FIG. 14. (a) Schematic level diagram of a weakly anharmonictransmon qubit subjected to a drive at transition frequencyωd = ωq. (b) Gaussian waveform with standard deviationσ. (c) Fourier transform of (b) showing how the short pulselengths lead to significant overlap with the ω1→2

q transition,separated from ωq by the anharmonicity α. (d) Waveformof a Xπ pulse without DRAG modulation. (e) Effect of thewaveform from (d) on a qubit initialized in the |0〉 state withα = −200 MHz and ωq = 4 GHz. The dephasing error isvisible as a deviation from the |1〉 after the pulse. (f) Wave-form of a Xπ pulse with DRAG modulation for a qubit withanharmonicity α = −200 MHz and DRAG parameter λ = 0.5to cancel dephasing errors (see text for details). (g) Effect ofthe waveform from (f) on the same qubit as (e). Calculatedusing mesolve in the software package QuTiP192.

gates using DRAG pulsing now routinely reach fidelitiesF1qb & 0.9965,67,198,201–204. Other techniques also existfor operating single-qubit gates in a spectrally crowdeddevice205,206.

31

E. The iSWAP two-qubit gate in tunable qubits

As briefly mentioned in Sec. IV C, single-qubit gatessupplemented with an entangling two-qubit gate canform the gate set required for universal quantum compu-tation. The two-qubit gates available in the transmon-like superconducting qubit architecture can roughly besplit into two broad families as outlined previously:one group requiring local magnetic fields to tune thetransition frequency of qubits and one group consist-ing of all-microwave control. There exist several hy-brid schemes that combine various aspects of these twocategories and, in particular, the notions of tunablecoupling and parametric driving are proving to be im-portant ingredients in modern superconducting qubitprocessors63,67,89,103,106,207–213. In this section, however,we start by introducing the iSWAP gate, and then reviewthe CPHASE (controlled-phase) in Section IV F and theCR (cross-resonance) in Section IV G. We briefly reviewa few other two-qubit gates and discuss their merits inSections IV G 4 and IV H.

1. Deriving the iSWAP unitary

As we saw in Sec. II, Eq.31 the interaction term be-tween two capacitively coupled qubits (in the two-levelapproximation) is given by

Hqq = gσy1 ⊗ σy2, (102)

where g is the coupling strength and ⊗ is used to em-phasize the tensor product. If the capacitive coupling ismediated through a bus resonator, then214,215

g → gq-r-q = g1g2(∆1 + ∆2)2∆1∆2

, (103)

where gi is the resonator coupling to qubit i (depen-dent on the qubit-resonator coupling capacitance Cqir)and ∆i = ωqi − ωr is the detuning of qubit i to the res-onator. In the simpler case where the qubits are directlycoupled216,

g → gq-q = 12√ωq1ωq2

Cq-q√Cq-q + C1

√Cq-q + C2

, (104)

where Cq-q is the qubit-qubit coupling capacitance andCi is the capacitance of qubit i. Throughout this sec-tion, we will assume a direct capacitive coupling betweenqubits of the flux-tunable transmon type, so that g = gq-qand ωqi → ωqi(Φi). For simplicity, we suppress the ex-plicit flux dependence of the ωqi’s and simply refer to thecoupling as g. Equation (102) can be rewritten as

Hqq = −g([σ+ − σ−]⊗ [σ+ − σ−]

), (105)

and then using the rotating wave approximation again(i.e. dropping fast rotating terms) we arrive at

Hqq = g(eiδω12tσ+σ− + e−iδω12tσ−σ+) , (106)

where we have introduced the notation δω12 = ωq1−ωq2and suppressed the explicit tensor product between qubitsubspaces. If we now change the flux of qubit 1 to bringit into resonance with qubit 2 (ωq1 = ωq2), then

Hqq = g(σ+σ− + σ−σ+) = g

2 (σxσx + σyσy) . (107)

The first part of Eq. (107) shows that a capacitive inter-action leads to a swapping of excitations between the twoqubits, giving rise to the ‘swap’ in iSWAP. Moreover, dueto the last part of Eq. (107), this capacitive coupling isalso sometimes said to give rise to an ‘XY ’ interaction217.The unitary corresponding to a XY (swap) interactionis

Uqq(t) = e−ig2 (σxσx+σyσy)t =

1 0 0 00 cos(gt) −i sin(gt) 00 −i sin(gt) cos(gt) 00 0 0 1

(108)

Since the qubits are tunable in frequency, we can nowconsider the effect of tuning the qubits into resonancefor a time t′ = π

2g ,

Uqq

(π

2g

)=

1 0 0 00 0 −i 00 −i 0 00 0 0 1

≡ iSWAP. (109)

From this result, we see that a capacitive coupling be-tween qubits turned on for a time t′ (inversely related tothe coupling strength in units of radial frequency) leadsto implementing a so called ‘iSWAP’ gate215,216,218–221,which acts to swap an excitation between the two qubits,and add a phase of i = eiπ/2. For completeness, we notethat for t′′ = π

4g the resulting unitary,

Uqq

(π

4g

)=

1 0 0 00 1/

√2 −i/√2 0

0 −i/√2 1/√

2 00 0 0 1

≡ √iSWAP, (110)

is typically referred to as the ‘squareroot-iSWAP’ gate.The

√iSWAP gate can be used to generate Bell-like su-

perposition states, e.g. |01〉+ i|10〉.To elucidate the operating principle behind an iSWAP

implementation we show the spectrum of a flux-tunable qubit using typical transmon-like parameters inFig. 15(a). The iSWAP is performed at the avoided cross-ing where Φ = ΦiSWAP. By preparing QB1 in state |1〉,moving into the avoided crossing, waiting there for a timeτ (see pulse-sequence in inset in Fig. 15(b)), the excita-tion is swapped back and forth between the two qubits,as shown in Fig. 15(b). In Fig. 15(c), we plot linecutsof (b) at ΦiSWAP, showing the excitation oscillating backand forth between |01〉 and |10〉 with the predicted timet′ = π/2g. In turn, the frequency of the oscillation canbe used to extract the strength of the coupling, 2

t′ = gπ .

32

(a)Ti

me,

τ

(c)(b)

p

initial pointfinal point

Freq

uenc

y (G

Hz)

Magnetic flux, qubit 1 (Φ0)

ProbabilityMagnetic flux, qubit 1 (Φ0)

ΦiSWAP

ΦiSWAP

5.0

1/2

2/2

3/2

4/2

0.10.0

0 0.5 1

0.2 0.3 0.4

4.5

X0

0fluxQ

B1 τ p p

FIG. 15. (a) Spectrum of two transmon qubits (written in thecombined basis as |QB1,QB2〉) as the local flux through theloop of qubit 1 is increased. Black/dashed lines with arrowsindicate a typical flux trajectory to demonstrate operation ofiSWAP gate. (b) Probability of swapping into the |01〉 stateas a function of time and flux. The pulse sequence correspondsto preparing |10〉 and performing a typical iSWAP operation(for a time τ). (c) Probabilities of |01〉 (black) and |10〉 (gray)at Φ = ΦiSWAP (white dashed line in (b)) as the time spentat the operating point (τ) is increased. This simulation doesnot include any decay effects.

So far we have ignored the role of the single-qubitphases acquired by tuning the qubit frequency. Refer-ring to the pulse-sequence shown in the top panel ofFig. 15(a), we see that each qubit will acquire a phasegiven by

θz =∫ τ

0dt (ωq − ω(t)) . (111)

This phase can be conveniently removed either by sub-sequent application of virtual-Z gates to all followingpulses191, or by shaping the waveform of the excursionsuch that single-qubit phases are exactly cancelled222.

Equations (104) and (108) together present a useful re-sult from a quantum processor design perspective: Theoperating regime, frequency and time τ of the iSWAPcan be calculated (typically simulated) to high precision,before any processor fabrication is undertaken. The only‘quantum’ parts that enter gqq (and gq-r-q) are the qubit

frequencies, ωq1(Φ1) and ωq2(Φ2). If the Josephson en-ergies of the qubits are known (which they typically are,from fabrication parameters), then by simulating the ca-pacitances in gqq or gq-r-q, the time τ and the pulseshapeneeded to implement an iSWAP can be estimated tohigh precision. Typical values of the coupling strength,g/(2π), for architectures using the iSWAP gate is 5-40MHz and are often very close to expectations from EMsimulations219,221–223.

2. Applications of the iSWAP gate

The iSWAP cannot generate a CNOT gate by itself.Rather, to implement a CNOT gate requires stringingtogether two iSWAPs and several single qubit gates217,

•=

Z-π2iSWAP

Xπ2

iSWAPXπ

2Zπ

2Zπ

2

(112)As evident from Eq. (112), the iSWAP gate in generalneeds to be used twice to generate a single CNOT, leadingto a significant overhead when compiling CNOT–densecircuits from iSWAP gates. However, depending on thecontext, the iSWAP can be used efficiently (i.e. with-out any two-qubit gate overhead) to mimic the behav-ior of a CNOT. Typically such circuits will not be com-pletely equivalent, but will share certain salient featuresfor specified input states. As an example of this proce-dure, Neeley et al.220 demonstrated the generation of a3-qubit Greenberger-Horne-Zeilinger (GHZ) state (whichrequires two subsequent CNOTs in the simplest construc-tion), by using only two iSWAPs in a circuit that cor-rectly generates the 3-qubit GHZ state on the |000〉 input.Moreover, the XY –interaction is a powerful tool for cer-tain types of quantum simulation algorithms224. If oneis interested in digital quantum simulation of spin–likesystems, then the XY –interaction can natively simulatee.g. a Heisenberg interaction,

HHeisenberg = Jxσxσx + Jyσyσy + Jzσzσz. (113)

This approach to the XY interaction was demonstratedby Salathe et al.222, where repeated application of theiSWAP gate interspersed with single-qubit rotations wasused to generate successive XY , XZ and Y Z interactionsthat lead to an aggregate HHeisenberg Hamiltonian. State-of-the-art operation of the iSWAP gate has also been usedto demonstrate a ten-qubit GHZ state225.

F. The CPHASE two-qubit gate in tunable qubits

In our discussion of the iSWAP gates, we assumed thatthe higher energy levels of the superconducting qubit donot play a role. As we show below, it turns out that for

33

the case of transmon qubits (with negative anharmonic-ity), the higher levels can in fact be utilized to generatea the CPHASE gate directly64,226.

Recall from Sec. IV C that the CPHASE gate imple-ments the following unitary,

UCPHASE =

1 0 0 00 1 0 00 0 1 00 0 0 −1

(114)

Our goal for the remainder of this section is to show thatthe unitary operator of the CPHASE gate appears natu-rally for capacitively coupled transmon superconductingqubits and review a few of the modern applications of thisgate. We have chosen to include a considerable amount ofdetails for the implementation of this gate, as a means toreview some of the issues one has to resolve, to engineerhigh quality two-qubit gates.

The structure of the matrix in Eq. (63) indicates thatwe need to apply a phase (−1 = eiπ) to the qubits when-ever both are in the excited state |11〉. Considering thenature of the XY interaction, which couples |01〉 ↔ |10〉and leads to the iSWAP gate (see previous section), weexpect avoided level crossings to exist between higher lev-els, e.g. |11〉 ↔ |20〉 and |11〉 ↔ |02〉. The flux-tunableimplementation of the CPHASE gate relies on this higher-level avoided crossing.

To motivate this intuition we plot the spectrum for twocoupled transmon qubits, in Fig. 16(a), including levelswith two excitations, as the local magnetic flux in qubit1 is being tuned. The Hamiltonian for this spectrum,written in the |00〉, |01〉, |10〉, |11〉, |02〉, |20〉-basis, is ap-proximately given by,

H2 excitations =

E00 0 0 0 0 00 E01 g 0 0 00 g E10 0 0 00 0 0 E11

√2g√

2g0 0 0

√2g E02 0

0 0 0√

2g 0 E20

,(115)

where Enm = Eq1n (Φ1)+Eq2

m (Φ2) and En(Φi) is the flux-dependent energy of the i-th level of a transmon52, andthe |02〉, |20〉 ↔ |11〉 transitions are scaled by a fac-tor√

2 due to the higher photon number. In Fig. 16,we plot the frequencies ωnm = Enm − E00 calculatedfrom Eq. (115), using standard, symmetric, transmon-like parameters, as the local magnetic field of qubit 1 isincreased.

The result of the higher levels on the computationalbasis can be understood by considering a concrete ex-ample. By preparing the combined qubit state |11〉 andmoving slowly towards the avoided crossing between |11〉and |20〉 at ΦCPHASE, waiting for some time τ and mov-ing back (see black line with arrows in Fig. 16(b)), theresulting unitary operator in the computational basis is

(a)

iSWAP

(τ)

(τ)

CPHASE

(b)

ω/ 2

π (G

Hz)

ω/ 2

π (G

Hz)

10

9

8

7

6

5

4

ΦiSWAPΦCPHASE

ΦCPHASE

0.10.0 0.2

0.20.15

0.3 0.4

Magnetic flux, qubit 1 (Φ0)

Magnetic flux, qubit 1 (Φ0)

ζ

initial pointfinal point

FIG. 16. (a) Spectrum of two coupled transmon qubits (usingtypical transmon-like values for Josephson energies and capac-itances) as the local magnetic flux for qubit 1 is varied. Thetwo lower branches corresponding to |01〉 and |10〉 are involvedin the iSWAP gate operation at Φ = ΦiSWAP. The avoidedcrossing indicated in the black rectangle is used to implementthe conditional phase gate (CPHASE), at Φ = ΦCPHASE. Blackline with arrows indicate a typical trajectory used to imple-ment a CPHASE gate (starting at the black circle and endingat the gray circle). (b) Zoom in of the |20〉 ↔ |11〉 avoidedcrossing highlighted in the black box in (a) at Φ = ΦCPHASE.The parameter ζ quantifies the difference in energy between|11〉 and |01〉+ |10〉 and `(τ) is the trajectory in (Φ, t)–space.

given by

Uad =

1 0 0 00 eiθ01(`) 0 00 0 eiθ10(`) 00 0 0 eiθ11(`)

, (116)

where

θij(`(τ)) =∫ τ

0dt ωij [`(t)] (117)

is the phase acquired by the state |ij〉 along the trajectory` in (Φ, t)-space during time τ . The movement should besufficiently slow on a time-scale set by g that the mov-ing state never populates the |20〉 state, i.e. the move-ment should be adiabatic. In terms of applied flux, theavoided crossing between the |11〉 ↔ |20〉 state happens

34

before |10〉 ↔ |01〉 (due to the negative anharmonicity ofthe transmons, α ≈ −Ec) and consequently ` does nottake the states through the ΦiSWAP operating point. Asshown in Fig. 16(b) we can define a parameter (typicallydenoted ζ) quantifying the difference in phase acquiredby the |11〉 relative to the single excitation states,

ζ = (ω11 − (ω01 + ω10)) . (118)

The parameter ζ can be thought of as the result (in thecomputational space) of the repulsion of |11〉 due to the|20〉 state. If we now choose a trajectory `π, designed sothat

∫ τ0 ζ(`π(t))dt = π, then∫ τ

0ζ(t)dt = π = θ11(`π)− (θ01(`π) + θ10(`π)) . (119)

Inserting this expression into Eq. (116) we see that

Uad =

1 0 0 00 eiθ01(`π) 0 00 0 eiθ10(`π) 00 0 0 ei(π+θ01(`π)+θ10(`π))

.(120)

After the adiabatic excursion, one can now apply single-qubit pulses (or use virtual-Z gates) to exactly cancel thesingle-qubit phases such that θ10(`π) = θ01(`π) = 0. Thischanges Uad to

Uad =

1 0 0 00 1 0 00 0 1 00 0 0 eiπ

=

1 0 0 00 1 0 00 0 1 00 0 0 −1

= UCPHASE. (121)

From Eq. (121) it is evident that an adiabatic movementof |11〉, followed by single-qubit gates (virtual or real)efficiently implements a CPHASE and, through Eq. (66),also efficiently implements a CNOT. The CPHASE gate isone of the workhorses of modern superconducting qubitprocessesors with gate fidelities & 0.9965,227.

One is, of course, free to choose an arbitrary trajec-tory `φ that implements the phase e−iφ on the |11〉 state.Assuming that the single-qubit phases are properly can-celled, one sees that the arbitrary phase version of theCPHASE gate (typically denoted CZφ) can be written as

CZφ =

1 0 0 00 1 0 00 0 1 00 0 0 e−iφ

= exp

[−iφ4 (σz ⊗ σz − σz ⊗ 1− 1⊗ σz)

]. (122)

Because of the form of Eq. (122), one can think of theavoided crossing with the higher levels outside the com-putational subspace as giving rise to an effective σz ⊗ σzcoupling within the computational subspace226.

An alternative to the adiabatic approach outlinedabove to realize CPHASE is to make a sudden excur-sion to the ΦCPHASE operating point, after waiting atime t = π/

√2g, the state will have completed a single

Larmor-type rotation from |11〉 to |02〉 and back again to|11〉, but in the process, acquired an overall π phase, sim-ilar to the iSWAP gate, but in the |11〉, |20〉 subspace54.In fact, such excursions near or through avoided crossingsleading to adiabatic and non-adiabatic transitions havebeen studied extensively in the context of interferometry,cooling, spectroscopy, and quantum control118,228–237.

The remainder of this subsection is devoted to anoverview of some of the recent advances and demonstra-tions using the CPHASE gate since its first demonstra-tion in 2009 where it was used to generate Bell-statesand demonstrate two-qubit algorithms64.

1. Trajectory design for the CPHASE gate

The (adiabatic) implemention of UCPHASE outlinedabove assumed that the trajectory `π was completelyadiabiatic and that the |11〉 state never left the compu-tational subspace. Since the fidelity of gates is boundedfrom above by the coherence times of the qubits, shortgate times are desirable238. This presents a tension foroptimally operating the CPHASE gate – fast operationin conjunction with the need for adiabatic operation. Arelevant question is then: what is the optimal trajectory`?π that implements the necessary phase as fast as pos-sible, with as little leakage as possible, for a given sizeof the avoided crossing between |11〉 and |20〉? Given atypical coupling rate g/2π ≈ 20 MHz (as discussed inSec. IV E), one expects a heuristic lower time limit to be2π/g ≈ 50 ns (stronger coupling of course leads to shortergate times, but will limit the on/off ratio of the gate).Traditional optimal control of adiabatic movement as-sumes the movement is through the avoided crossing (seee.g. Ref. 239), but the trajectory `π moves close to andthen back from the avoided crossing. This modificationto the adiabatic movement protocol was addressed byMartinis and Geller240, specifically in the context of er-rors for a CPHASE gate implemention. The authors showthat non-adiabatic errors can be minimal for gate timesonly slightly longer than 2π/g using an optimal wave-form (based on a Slepian waveform241) to parametrizethe trajectory `?π(τ).

2. The CPHASE gate for quantum error correction

Using the approach of Martinis and Geller, Barendset al. were able to demonstrate a two-qubit gate fidelityFCPHASE = 0.9944 (determined via a technique knownas ‘interleaved randomized benchmarking’242–245). Thisimplementation had a gate time τ = 43 ns and was imple-mented with the `?π waveform65, in an “xmon” device85

– a transmon with a “+”-shaped capacitor. A two-qubit

35

gate fidelity F > 0.99 represents a significant milestone,not just from a technical and engineering perspective,but also from a foundational standpoint: The surfacecode (a quantum error correcting code) has a lenientfault-tolerance threshold of ∼ 1%246–248. This means,roughly speaking, that if the underlying operations onthe qubits have fidelities F > 0.99, then by addingmore qubits to the circuit (and correctly implementingthe fault-tolerant quantum error correction protocol) theoverall error-rate can be reduced, and one can in principleperform arbitrarily long quantum computations, with-out errors spreading uncontrollably and corrupting thecalculation. Because of its relatively lenient thresholdunder circuit noise (compared to e.g., Steane or Shorcodes177,249,250) and its use of solely nearest-neighborcoupling, the surface code is one of the most promisingquantum error correction codes for medium-to-large scalequantum computing in solid state systems246. Therefore,surpassing the fault-tolerance threshold using CPHASErepresents a significant milestone for the field251. More-over, practical blueprints for implementing scalable sub-cells of the surface code using the CPHASE as the funda-mental two-qubit gate have also been proposed71 as wellas in-situ calibration protocols for large-scale systems op-erating with CPHASE252. For a full review of the pros andcons of various quantum error correcting codes we referthe interested reader to e.g. an introductory review arti-cle Ref. 253, or any of the excellent textbooks and moredetailed review articles in Refs. 177, 179, 250, 253–256.

Returning to the CPHASE gate, numerical optimiza-tion of `?π was demonstrated by Kelly et al.227 using theinterleaved randomized benchmarking sequence fidelityas a cost function to push a native implementation of`?π with a fidelity F = 0.984 up to F = 0.993, sur-passing the surface code fault tolerance threshold. Inthe same work that demonstrated FCPHASE = 0.9944,Barends et al.65 used the CPHASE gate to generateGHZ states, |GHZ〉 =

(|0〉⊗N + |1〉⊗N

)/√

2, of up toN = 5 qubits, with a fidelity for the N = 5 state ofF = Tr (ρidealρN=5) = 0.817. The protocol for generat-ing the GHZ state with N = 2 and N = 3 from CPHASEwas originally demonstrated by DiCarlo et al.54,64. Thetextbook route to generating the N = 2 GHZ state, |Φ+〉(a Bell state) from the all-zero input is

|0〉 H •

|0〉

= |Φ+〉(123)

An equivalent circuit using CPHASE and native single-qubit gates in superconducting qubits is:

|0〉 Yπ/2 •

|0〉 Y−π/2 • Yπ/2

. (124)

By repeating the operation inside the dashed boxon additional qubits, an N -qubit GHZ state can begenerated65. Since the demonstration of the N = 5 GHZ

state using the CPHASE gate, the gate has been deployedto demonstrate several important aspects of quantum in-formation processing using superconducting qubits. Anine-qubit implementation of the five-qubit repetitioncode (five data qubits + four syndrome qubits)253 wasdemonstrated, and the error suppression factor of a sin-gle logical quantum bit was shown to increase as the en-coding was changed from three data qubits to five dataqubits66. Similarly, in a five qubit processor the three-qubit repetition code with artificially injected errors wasdemonstrated257, building on earlier results utilizing acombination of iSWAP and CPHASE gates to performparallelized stabilizer readout258.

3. Quantum simulation and algorithm demonstrationsusing CPHASE

As an example of the utility of the CPHASE gate, webriefly discuss a particular demonstration of a digitalquantum simulation. In this context, the CPHASE gatehas been utilized to simulate a two-site Hubbard modelwith four fermionic modes, using four qubits259. Usingthe Jordan-Wigner transformation260,261, it is possible tomap fermionic operators onto Pauli spin matrices262. Asshown in Ref.259 a Hubbard model with two fermionicmodes, whose Hamiltonian is given by

HHubbard, two mode = −t(b†1b2 + b†2b1) + Ub†1b1b†2b2 (125)

can be written in terms of Pauli operators as,

H = t

2 (σx ⊗ σx + σy ⊗ σy) (126)

+ U

4 (σz ⊗ σz + σz ⊗ 1+ 1⊗ σz) , (127)

where U is the repulsion energy and t is the hoppingstrength. Similar to the Heisenberg interaction discussedbriefly in Sec. IV E, it is now a question of producingασi ⊗ σi-type interactions, where the prefactor α can betuned. Using the CZφ version of CPHASE, a UZZ(φ) =exp

(−iφ2σz ⊗ σz

)unitary can be generated via

UZZ(φ) =Xπ

CZφAπ

CZφAπ Xπ

,

(128)where Aπ ∈ Xπ,Yπ is used to allow for small and neg-ative angles. Finally, for completeness, we mention analternative approach to creating UZZ, given by42,263

UZZ(φ) =• •

Zφ/2

,(129)

which has the benefit of relying on CPHASE (throughthe CNOTs), and the angle can be controlled using the

36

single-qubit Z gates. We refer the interested reader totwo reviews on quantum simulations, see e.g. Refs. 264and 265.

The CPHASE gate has also been used in a vari-ety of other contexts, e.g., for calculating the dissoci-ation of diatomic hydrogen (H2) using the variationalquantum eigensolver method266, for feed-forward basedteleportation experiments267,268, as well as initial stepstowards demonstrating quantum supremacy269 and a2 × 2 implementation of the Harrow-Hassidim-Lloydalgorithm270,271. In the field of hybrid semiconductingnanowire/superconducting qubits (known as the “gate-mon” approach272–274), where the qubit frequency ismodified by electrostatically changing the density of car-riers in a semiconducting region with proximity-inducedsuperconductivity, the CPHASE gate was also demon-strated between two nanowire qubits275.

One may worry that operating a qubit by moving itsfrequency can lead to overlap with frequencies alreadyused by other qubits, in a system with multiple qubits.This issue is known as frequency crowding. While theuse of asymmetric transmons (with two sweet spots inthe range [−Φ0,+Φ0], recall Fig. 2(c)) may help allevi-ate some frequency crowding issues, a more long-termstrategy is needed. One way to circumvent the problemis to utilize on/off tunable coupling schemes, in whichqubits can exchange energy only if a coupler activatesthe interaction63,103. To address this issue in the contextof the CPHASE gate, Chen et al.103 demonstrated a de-vice (named “the gmon”) where the qubit interaction canbe tuned with an on/off ratio on the order of 1000, and aCPHASE gate fidelity of F = 0.9907 was demonstrated.

This concludes the introduction to the physics and op-eration of the CPHASE gate in its native form. In theremainder of this section we will introduce a few of themicrowave-only gates that have been demonstrated in aneffort to sidestep the need for local tunability (and theresulting increased sensitivity to noise) as required by theiSWAP and CPHASE gate.

G. Two-qubit gates using only microwaves

One common (potential) drawback for the iSWAPand CPHASE gates is that their operation requires flux-tunable qubits. Introducing a new control knob, such asflux control, in turn also introduces a new noise channelfor the system. Furthermore, the need for flux-tunabilityincreases the sensitivity of the devices to flux noise bytuning the qubits from their “sweet spots”, increases thedephasing rate. From this perspective, one could envi-sion using all-microwave-based gates to remedy these is-sues. To this end, the cross-resonance (“CR”) gate wasdeveloped for operating fixed-frequency superconductingqubits276–278, which typically feature longer lifetimes andreduced sensitivity to flux noise.

1. The operational principle of the CR gate

To elucidate the operation of the CR gate, we brieflyrevisit the driving Hamiltonian derived in Sec. IV D.There, we considered only a single qubit. However, ifone extends this formalism to two qubits, see Fig. 17(a)denoting the frequency difference by ∆12 = ωq1−ωq2 andthe coupling by g ∆12, and performing a Schrieffer-Wolff transformation to go to the dressed state picture,the driving Hamiltonians for qubit 1 and 2 become277,279

Hd,1 = ΩVd1(t)(σx ⊗ 1+ ν−1 1⊗ σx + µ−1 σz ⊗ σx

)(130)

Hd,2 = ΩVd2(t)(1⊗ σx + ν+

2 σx ⊗ 1+ µ+2 σx ⊗ σz

)(131)

where

µ±i = ± g

∆12

αi(αi ∓∆12) (132)

ν±i = ± g

∆12

∓∆12

(αi ∓∆12) (133)

and ΩVdi(t) is the driving for qubit i. From Eq. (130),it is evident that if we drive qubit 1 at the frequency ofqubit 2, then to qubit 2, this will look like a combinationof ν−1 1 ⊗ σx and µ−1 σz ⊗ σx. This means that the Rabioscillations of qubit 2 will have a frequency given by

ΩRabiQB2 = ΩVd1

(ν−1 + z1µ

−1), (134)

where z1 = 〈σz1〉, and z1 depends on the state of qubit1. This effect is demonstrated in Fig. 17(c), where asimulated drive is applied to qubit 1 while the resultingRabi oscillations in qubit 2 are recorded. We have usedtypical fixed-frequency transmon parameters from exper-iments, and we have included a spurious cross-talk termη = 0.03.245,280. In Fig. 17(d), we plot the difference inangle in the (z, y) plane acquired by qubit 2 for differentinitializations of qubit 1, ∆φ = φzy|00〉 − φzy|10〉. For thisparticular choice of parameters, the cross-resonance gateachieves a π-phase shift in ≈ 200 ns.

This strategy was first demonstrated using flux-tunable transmons in Ref.281, where a Bell state withfidelity Fbell = 〈Φ+|ρ|Φ+〉 = 0.90 was achieved. Usingquantum process tomography the gate fidelity was foundto be FQPT = 0.81. By moving to fixed-frequency qubitswith increased lifetimes, the gate fidelity was increasedto FQPT = 0.98 (with subtraction of state initializationand measurement errors)280. For completeness, we notethat due to the form of the last term in Eq. (130), theCR gate is also sometimes denoted the ZXθ gate. Theunitary matrix representaion of the CRθ gate is

UCRθ = e−i2 θσz⊗σx

=

cos θ2 −i sin θ

2 0 0−i sin θ

2 cos θ2 0 00 0 cos θ2 i sin θ

20 0 i sin θ

2 cos θ2

(135)

37

(a)

(d)(c)1

0

-1

1

0

-1an

gle,

QB2

(rad

)

QB2QB1

0 200100

coupler

g (b)

~

ω2~

~ω1 ω2 g2-2Δ12

g2

2Δ12+

QB1=

QB1=

Expe

ctat

ion

valu

e, Q

B2

Time (ns)0

0

π

200100Time (ns)

FIG. 17. (a) Schematic circuit diagram of two fixed frequencytransmons coupled through a resonator yielding an overallcoupling coefficient g. Qubit 1 driven at the frequency ofqubit 2 leads to the CR gate. (b) Schematic level diagramof the always-on coupling leading to dressed states |01〉 and|10〉 with ∆12 = ω1 − ω2. (c) Simulations of the expectationvalues of 〈σz〉 and 〈σy〉 for qubit 2 as a drive at the frequencyof qubit 2 is applied to qubit 1. Upper panel shows regularRabi oscillations when qubit 1 is in the |0〉 state. Bottompanel shows a modified Rabi frequency when qubit 1 is in |1〉state, in accordance with Eq. (134). (d) Difference in anglein the (z, y) plane as a function of length of the applied driveto qubit 1. At approximately 200 ns π-phase shift has beenacquired.

where θ = −µ−1 ΩVd1(t), which can be used to generate aCNOT with the addition of only single-qubit gates,

•=

Zπ2

CR−π2

Xπ2 •

,(136)

up to a phase eiπ/4.

2. Improvements to the CR gate and quantum errorcorrection experiments using CR

Since qubit 1 is being driven off-resonance, an ac-Starkshift will add a term ∝ σz1 to the driving Hamiltonian ofqubit 1. The effect of both the spurious ac Stark shift andthe direct ν−1 1σx single-qubit rotations was studied inRef.245. By modifying the original CR protocol to effec-tively “echo away” the two unwanted contributions fromthe σz1 and 1σx terms, the fidelity of the CR gate wasimproved to FCR = 0.8799245, using quantum processtomography. Using interleaved randomized benchmark-ing of this improved “echo-CR”-gate (eCR−π2 ), a gate

fidelity of FeCR−π2= 0.9347 was achieved. This gate

implementation was used to demonstrate two-qubit par-ity measurements in a three-qubit device282, as well asdetecting bit-flip and phase-flip errors in a Bell state en-coded in a four-qubit device283, with gate fidelities frominterleaved randomized benchmarking in the range 0.94to 0.96. Using a similar device, but with five qubits,weight-four parity measurement of the forms ZZZZ andXXXX were demonstrated284, where the crosstalk toqubits not involved in the CR gates was studied, leadingto the development of a four pulse eCR4-pulse scheme.

Based on improvements in the analysis of the Hamil-tonian describing the CR drive, Sheldon et al.202 sub-sequently demonstrated a version of the CR which re-duced the gate time to τ = 160 ns and added an ac-tive cancellation tone to the eCR previously developed.Using this “active cancellation echo CR” (aceCR), thefidelity was increased to FaceCR−π2

= 0.991, measuredwith interleaved randomized benchmarking. The samesequence without active cancellation on the same qubitsyielded FeCR−π2

= 0.948. The interested reader may con-sult the followup theoretical work285 with more detailson the effective Hamiltonian models. Other approachesto fast, high-fidelity cross-resonance gates have also beenproposed286. This series of improvements to the originalcross-resonance implementation has increased the gate fi-delity to beyond the threshold for fault-tolerance in a sur-face code, with similar quality to the CPHASE gate. Al-though improvements should still be made, with the ad-vent of the CR gate, superconducting qubit based quan-tum computing platforms now offer two entangling two-qubit gates that can be used for implementing surface-code based error correction schemes.

In the initial experiments using CR gates, the gatetimes were significantly longer than the typical CPHASEgate times (τCPHASE = 30 – 60 ns and τCR = 300 – 400ns), which to a large extent accounts for the observedCR gate fidelities. The time scale for CR operation is setby the frequency detuning, the anharmonicity, and thecoupling strength, through Eq. (132). This has the un-fortunate drawback that if qubits do not have intendedfrequencies (due to fabrication variation), it will be im-mediately manifest as longer gate times, and in turn, re-duced gate fidelity. As fabrication techniques are becom-ing more sophisticated and reliable, this problem may beof reduced importance. However, since the coupling inthe CR scheme is always on, there is an inherent tensionbetween well-isolated qubits for high-fidelity single-qubitoperations, and coupling qubits, for fast/high-fidelity twoqubit gates.

3. Quantum simulation and algorithm demonstrationswith the CR gate

Since the form of the CR Hamiltonian (σz ⊗ σx) isnot a (σx ⊗ σx + σy ⊗ σy)-type interaction (leading to

38

iSWAP gate) nor is it an the effective (σz ⊗ σz)-type(leading to CPHASE gate), one could question its appli-cability to quantum-simulation-type experiments, whichoften involves terms of the form σi⊗σi. However, by de-veloping a variational quantum eigensolver routine thatefficiently generates entangled trial states using just theCR interaction, Kandala et al.287 calculated the ground-state energy for H2, LiH, and BeH2. This experiment wasperformed on six fixed-frequency qubits, and it employeda technique for compact encoding of the Hamiltonianscorresponding to each molecule288. As of this writing,this experiment represents the largest molecule for whichthe ground state has been found using a purely quantumprocessing approach.

The CR gate is also the native two-qubit gate availableon the IBM Quantum Experience quantum processor289,which is accesible online. Using the IBM Quantum Expe-rience processor, Takita et al.290 demonstrated an imple-mentation of a two-logical-qubit (four physical qubit) er-ror detection code291. The implementation was inspiredby the proposal of Gottesman292, which proposed a min-imal experiment to claim observation of fault-tolerantencodings254, using a four qubit error detection code ina five qubit setup. Due to constraints on the connec-tivity, the work by Takita et al. demonstrated a mod-ified version of the Gottesman encoding, in which twological qubits are initialized, but only one of them in afault-tolerant manner. By artificially injecting an errorin the state preparation circuit, the authors demonstratethat the probability of correctly preparing a fault tolerantstate is greater than the probability of correctly prepar-ing a non-fault-tolerant qubit. This behavior is consistentwith expectations for how fault-tolerant encodings work.Simultaneously, Vuillot293 also used the IBM QuantumExperience machine to study fault-tolerant schemes en-coded in that connectivity.

Beyond the applications to error-correction and error-detection, the cross-resonance gate has also been em-ployed in early demonstrations of quantum advantagesin machine learning. Riste et al.294 studied the so-called“learning parity with noise” problem, in which one at-tempts to learn a bit-string k by querying an oracle func-tion f(D, k) = D · k mod 2 with a user-input bit-stringD. In a first implementation of this problem, the authorsshow that for a specific instance of the bit-string k = 11,a learner with access to quantum operations needs fewerqueries to the function f . However, by extending themodel of learning parity with noise, the authors demon-strated a consistent advantage of the learner with accessto quantum operations294.

The CR gate was also used to demonstrate the imple-mentation of a supervised learning algorithm where thefeature space is encoded as quantum data on the Blochsphere263. In typical supervised learning, an algorithmis exposed to a training set of labeled data, and is subse-quently asked to classify a new, unlabeled set of data295.In the support vector machine (SVM) approach to suchproblems, the data is then mapped non-linearly onto the

so-called “feature space”, in which the trained algorithmhas constructed a separating hyperplane to classify thedata. While a full “quantum Support Vector Machine”proposal exists, the algorithm assumes that the data arealready present in a coherent superposition296. Instead,Havlicek et al.263 proposed, and demonstrated, that map-ping the classical data non-linearly onto the Bloch spherecan also be utilized to provide a quantum advantage.For a wider discussion of the important role of quantumdata in many quantum machine learning algorithms, thereader is referred to Ref.297

4. Other microwave-only gates: bSWAP, MAP, and RIP

The CR gate (as outlined above) is not the only all-microwave two-qubit gate available. In particular, thebSWAP gate298 is an interesting alternative. The bSWAPgate directly drives the |00〉 ↔ |11〉 transition, made pos-sible by interactions with the higher levels of the qubit,see Fig. 18. Usually, the matrix element for such atransition is small (3rd order in the coupling strength),but if the detuning between the qubits is equal to theanharmonicity, the transition rate is enhanced. Apply-ing a sequence of Schrieffer-Wolff transformations to thecoupled-qubit system, and using a carefully chosen drivefrequency (close to the midpoint of ωq1 and ωq2), it canbe shown279 that the drive gives rise to a unitary operator

U = UbSWAP(θ, φ)UZZUIZ−ZI (137)

with

UbSWAP(θ, φ) =

cos θ 0 0 −ie−i2φ sin θ

0 1 0 00 0 1 0

−ie−i2φ sin θ 0 0 cos θ

,(138)

The two unitaries UZZ and UIZ−ZI only contain termsthat commute with UbSWAP(θ, φ), and their effects can beoffset in post-processing279. In Eq. (138), φ is the phaseof the drive relative to the single-qubit drive pulses, andθ = ΩBt with

ΩB =−2gΩ2 (−gγαΣ + γ2α2(α1 + ∆12) + α1(α2 −∆12)

)(α1 + ∆12)(α2 −∆12)∆2

12,

(139)where Ω is the amplitude of the drive, γ is a dimen-sionless parameter quantifying the coupling coefficientof the drive to qubit 2 in units of coupling strengthto qubit 1, and αΣ = α1 + α2. Explicit derivationsleading to Eq. (137) can be found in the supplementof Ref.298. By applying UbSWAP for a time that yieldsθ = π/2, and with φ = 0, the resulting gate is denotedbSWAP and can act as the entangling gate (together withsingle-qubit gates) that forms a universal gate set. More-over, the power of the bSWAP becomes apparent whenone applies it for the time that yields θ = π/4, which

39

Eb

Ea

computational subspace

FIG. 18. Schematic of the level structure of two coupledqubits (including higher levels) with indication of the tran-sitions utilized in the iSWAP, bSWAP, CPHASE and MAPgate. See text for details. Figure inspired from Ref. 70.

from the ground state |00〉 directly produces the entan-gled Bell state |00〉 + eiφ|11〉. In line with the defini-tion of

√iSWAP, this gate is denoted the

√bSWAP. In

the work by Poletto et al.298, the fidelity of the bSWAPgate was FbSWAP = 0.9 (determined from quantum pro-cess tomography). The main source of error was the in-creased dephasing during the relatively long high-powerpulse needed to drive the |00〉 ↔ |11〉 transition. ThebSWAP gate can be viewed as the superconducting qubitanalogue of the Mølmer–Sørensen gate299. In Fig. 18, weoutline the level diagram of two coupled qubits, alongwith the higher levels of the qubits. The arrows indicatewhich coupled states are utilized to implement the cor-responding gate. As an application of the bSWAP gate,Colless et al.300 used this gate to calculate energies ofthe excited states of a H2 molecule using a two-qubittransmon processor300.

Another all-microwave gate is the so-called”microwave-activated CPHASE” (or ‘MAP’ for short)70.The MAP gate is in spirit similar to the CPHASE gate,where noncomputational states are used to impact aconditional phase inside the computational subspace.In contrast to CPHASE, the MAP gate is implementedwithout tuning individual qubit frequencies. Rather,the canonical implementation of this gate comprises twofixed-frequency qubits, where the frequencies are care-fully designed (and fabricated), such that the |12〉 and|03〉 levels are resonant. This leads to a splitting of theotherwise degenerate |02〉 ↔ |01〉, and |12〉 ↔ |11〉 tran-sitions. By driving near resonance with the |n2〉 ↔ |n1〉transition, an effective σz ⊗ σz interaction is generated.In a setup comprising two fixed-frequency qubits, theMAP gate was used to implement the unitary

UMAP = exp[−iπ4 σz ⊗ σz

], (140)

with a gate fidelity FMAP = 0.87 (determined via quan-tum process tomography) in a time τMAP = 514 ns70.As the number of qubits in a system increase, one draw-back of this gate is the need for a precise matching ofhigher energy levels across multiple qubits, while simul-taneously avoiding spurious couplings to other modes inthe system.

The CR, bSWAP and MAP gates all have quite strin-gent requirements on the spectral landscape of the qubitsin order to obtain fast, efficient gate operation. Toaddress this issue, another all-microwave gate was de-veloped, the so-called “resonator induced phase gate”(“RIP”)301,302. The RIP gate operates by coupling twofixed-frequency qubits to a bus cavity, from which theyare far detuned. By adiabatically applying and removingan off-resonant pulse to the cavity, the system undergoesa closed loop in phase space, after which the cavity isleft unchanged, but the qubits acquire a state-dependentphase. By a careful choice of the amplitude and detuningof the pulse, and taking into account the dispersive shiftof the cavity, a CPHASE gate can be implemented on thetwo qubits. This effect was experimentally demonstratedby Paik et al.303 in a 3D transmon system55, where fourqubits are coupled to the same bus. In this setup, theRIP gate operation results in unitaries with weight on allfour qubits simultaenously. In order to isolate just thedesired two-qubit coupling terms, Paik et al. developeda “refocused” RIP (rRIP) gate that implements

UrRIP = exp[−iθσz ⊗ σzt

], (141)

where the coupling rate (for an unmodulated drive) scalesas

θ ∝(|ΩVd|2∆cd

)2

︸ ︷︷ ︸n

χ

∆cd, (142)

where n denotes the average number of photons in thebus, χ is the dispersive shift, and ∆cd is the detuning ofthe drive (d) from the cavity (c). By choosing θt = π/4, itis possible to implement the CPHASE gate. The power ofthe RIP gate lies in its capability to accommodate largedifferences in qubit frequencies. To demonstrate this,Paik et al.303 performed two-qubit randomized bench-marking between pairs of qubits in a four-qubit devicewith frequency differences spanning 0.38 GHz to 1.8 GHz,all with fidelities in the range 0.96-0.98 and gate times inthe range 285 to 760 ns.

H. Gate implementations with tunable coupling

Finally, we briefly review tunable coupling architec-tures, which have recently emerged as a promising al-ternative. The idea is to engineer an effective qubit-qubit coupling g that is tunable (typically by apply-ing a flux), and such gates are referred to as paramet-ric gates. This can be implemented in two different

40

ways: (i) The coupling strength between two qubits istuned by a flux, g → g(Φ(t))198,207,304–306, or (ii) theresonant frequency of the coupling element is modifiedωcoupler → ωcoupler(Φ(t))89,106,307–311, with a fixed g,leading to an effective time-dependent coupling param-eter. When the tunable coupling element is driven atfrequencies corresponding to the detuning of the qubitsfrom the coupler, an entangling gate can be implemented.

In a setup of type (ii), an implementation of theiSWAP gate was demonstrated by parametrically driv-ing a flux-tunable coupler between two fixed-frequencyqubits63, yielding a fidelity FiSWAP = 0.9823 (using inter-leaved randomized benchmarking) in a time τ = 183 ns.Similarly, the bSWAP (and iSWAP) gates were recentlydemonstrated, using a flux-tunable transmon connect-ing two fixed-frequency transmons. Driving the fluxthrough the tunable qubit at the sum frequency of thefixed-frequency transmons results in the bSWAP104 gate.This parametrically driven approach is generally sig-nificantly faster than implementations relying solely onfixed-frequency qubits.

A hybrid approach, in which a combination of tunableand fixed-frequency qubits is used, was recently demon-strated for both iSWAP and CPHASE gates67,105,211.This scheme has no added tunable qubits (or resonators)acting as the coupling element, but rather, relies solelyon an always-on capacitive coupling between the qubits,and the effective coupling is roughly half that of thealways-on coupling. The operational principle here isto modulate the frequency of the tunable qubit (usinglocal flux control) at the transition frequency correpond-ing to |01〉 ↔ |10〉 for iSWAP and |11〉 ↔ |02〉 forCPHASE. Using interleaved randomized benchmarking,the authors demonstrated FiSWAP = 0.94 (τ = 150 ns),and F02

CPHASE = 0.93 (τ = 210 ns) and F20CPHASE = 0.88

(τ = 290 ns), showing a slight asymmetry in the direc-tion in which the CPHASE is applied. This hybrid tech-nique was used in Ref.67 to demonstrate a four-qubitGHZ state with fidelity F4 qubit GHZ = 0.79 (using statetomography). Finally, this gate-architecture was used todemonstrate a hybrid quantum/classical implementationof an unsupervised learning task (determining clusteringof data), using nineteen qubits and supplemented by aclassical computer as part of the minimization loop312.

V. QUBIT READOUT

The ability to perform fast and reliable (high fidelity)readout of the qubit states is an important cornerstoneof any quantum processor3.

In this section, we give a brief introduction to howreadout is performed on superconducting qubits. Westart by reviewing the fundamental theory behind disper-sive readout – the most common readout technique usedtoday in the circuit QED architecture – in which eachqubit is coupled to a readout resonator. In the dispersiveregime, i.e. when the qubit is detuned from the resonator

frequency, the qubit induces a state-dependent frequencyshift of the resonator from which the qubit state can beinferred by interrogating the resonator.

Dispersive readout allows us to map the quantum de-gree of freedom of the qubit onto the classical response ofthe linear resonator, thus transforming the readout opti-mization process into obtaining the best signal-to-noiseratio (SNR) of the microwave signal used to probe theresonator.

We then provide guidance on how to optimize systemparameters to perform high-fidelity, single-shot readout.After choosing parameters, such as resonant frequenciesand coupling rates, we address the filter and amplifiercircuitry positioned in-between the qubit plane and thedata aquisition hardware outside of the dilution refrid-erator. On this note, we review the basic principles ofPurcell filters as well as parametric amplifiers, both ofwhich are necessary to obtain fast, high-fidelity readoutin scaled-up quantum processors.

A. Dispersive readout

A quantum measurement can be described as anentanglement of the qubit degree of freedom with a“pointer variable” of a measurement probe with a quan-tum Hamiltonian313, followed by classical measurementof the probe. In circuit QED, the qubit (the quantum sys-tem) is entangled with an observable of a superconduct-ing resonator (the probe), see Fig. 19(a), allowing us togain information about the qubit state by interrogatingthe resonator – rather than directly interacting with thequbit. Therefore, the optimization of the readout per-formance is translated to maximizing the signal-to-noiseratio of a microwave probe tone sent to the resonator,while minimizing the unwanted back-action on the qubit.

The qubit-resonator interaction is described by theJaynes-Cummings Hamiltonian314–316, previously intro-duced in Sec. II,

HJC = ωr

(a†a+ 1

2

)+ ωq

2 σz + g(σ+a+ σ−a

†) , (143)

where ωr and ωq denote the resonator and qubit frequen-cies, respectively, and g is the transverse qubit-resonatorcoupling rate. The operators σ+ and σ− represent theprocesses of exciting and de-exciting the qubit, respec-tively.

In the limit when the detuning between the qubit andthe resonator is small compared with their coupling rate,i.e. ∆ = |ωq − ωr| g, the energy levels of the twosystems hybridize and a vacuum Rabi splitting of fre-quency

√ng/π opens up, where n = 1, 2, 3... denotes the

resonator mode. In this regime, excitations are coher-ently swapped between the two systems. Although use-ful for certain two-qubit gate operations, recall Sec. IV E,such transverse interactions change the qubit state (sinceenergy is directly exchanged between the resonator and

41

Qubit

CHIP

10 mK3 K300 K

Res.

CRYOSTAT

SIGNAL DETECTIONSIGNAL CREATION

Purcell filter

Para. Amp.

AWG

(a)

00.0

Frequency, ωRF - ωr (a.u)

-

-2

0.5

1.0

0 1 2-1-2

(b)

|S11

| (a.

u.)

(c)

κ/2

2χ/2

θ (r

ad)

|S11|θ

gκ

ωRF ωLO

ωIF

FIG. 19. (a) Simplified schematic of a representative exper-imental setup used for dispersive qubit readout. The res-onator probe tone is generated, shaped and timed using anarbitrary waveform generator (AWG), and sent down into thecryostat. The reflected signal S11 is amplified, first in a para-metric amplifier and then in a low-noise HEMT amplifier,before it is downconverted using heterodyne mixing and fi-nally sampled in a digitizer. (b) Reflected magnitude |S11|and phase θ response of the resonator with linewidth κ, whenthe qubit is in its ground state |0〉 (blue) and excited state |1〉(red), separated with a frequency 2χ/2π. (c) Correspondingcomplex plane representation, where each point is composedof the in-plane Re[S11] and quadrature Im[S11] components.The highest state discrimination is obtained when probing theresonator just in-between the two resonances, (dashed line in(b)), thus maximizing the distance between the states.

the qubit) and is therefore not desired in the context ofquantum non-demolition (QND) readout, in which theoutcome of the quantum measurement is not altered inthe act of reading out the system.

In the dispersive limit, i.e., when the qubit is far de-

tuned from the resonator compared with their couplingrate g and the resonator linewidth κ, ∆ g, κ, thereis no longer a direct exchange of energy between thetwo systems. Instead, the qubit and resonator pusheach others’ frequencies. To see this, the Hamiltoniancan be approximated using second-order perturbationtheory214,317 in terms of g/∆, taken in the limit of fewphotons in the resonator. This is known as the disper-sive approximation, after which the Hamiltonian takesthe form

Hdisp =(ωr + χσz

)(a†a+ 1

2

)+ ωq

2 σz, (144)

where χ = g2/∆ is the qubit-state dependent frequencyshift, a so-called dispersive shift, see Fig. 19(b), allow-ing us to distinguish the two qubit states. This is anasymptotically longitudinal interaction, yielding a QNDmeasurement. Note that, in addition, the qubit fre-quency also picks up a Lamb shift, ωq = ωq + g2/∆, in-duced by the vacuum fluctuations in the resonator. Alsonote that the dispersive Hamiltonian in Eq. (144) is de-rived for a two-level atom¶. Taking the second excitedstate into account and introducing the anharmonicityα = ω1→2

q − ω0→1q modifies the expression for the dis-

persive shift:

χ = χ01 + χ12

2 = −g201∆

(1

1 + ∆/α

), (145)

which for a transmon qubit with α < 0 implies that thedispersive shift will depend on the detuning. This ef-fect is plotted in Fig. 20(a), where the second energylevel manifests itself as a second vertical asymptote at∆/2π = EC/h. It is also worth noting that for qubitmodalities with positive anharmonicity, e.g. flux qubits,the dispersive shift will also shift the sign62.

In the small photon-number limit, the interaction termof the Hamiltonian in Eq. (144) commutes with the qubitobservable‖, σz, resulting in a QND measurement313.This is an important condition for many applications inquantum information processing.

In the case when the resonator photon number n = a†aexceeds a critical photon number nc ≡ ∆2/(4g2), the dis-persive Hamiltonian in Eq. (144) is no longer a validapproximation214,318,319. Therefore, the critical pho-ton number sets an upper bound for the power levelof the resonator probe signal to maintain (an approx-

¶In reality, superconducting qubits, just like natural atoms, havehigher energy levels. These higher levels are outside of the com-putational subspace, but need to be taken into account for mostqubit simulations to get accurate predictions of frequency shifts.

‖This commutation is approximate and has an asyptotic dependenceon the qubit-resonator detuning

42

-0.4 -0.2 0.0 0.2 0.4-0.2

-0.1

0.0

0.1

0.2

Straddling Straddling regimeregimeregime

-1.0 -0.5 0.0-1.5-2.0-10

-8

-6

0

-2

-4

(a)

(b)

Dis

p. S

hif

t, ,χ

/2 (

GHz)

Dis

p. S

hif

t, ,χ

/2 (

MHz

)

Frequency detuning, /2 (GHz)

/2 = 0 /2 = EC/h

g/2 = 50 MHzg/2 = 100 MHz

g/2 = 50 MHzg/2 = 100 MHz

Frequency detuning, /2 (GHz)

FIG. 20. (a) Dispersive frequency shift χ/2π as a functionof qubit-resonator detuning ∆/2π, according to Eq. (145),for a transmon qubit with anharmonicity α/2π = −EC/h =−300 MHz, for qubit-resonator coupling rates g/2π = 50 MHz(blue) and g/2π = 100 MHz (red). The two vertical asymp-totes at ∆/2π = 0 and ∆/2π = EC divides the dispersiveshift into three regimes; For ∆/2π < 0, and ∆/2π > EC/h,the dispersive shift is negative and χ/2π → 0− as ∆→ ±∞.For 0 < ∆/2π < EC/2π, the dispersive shift χ/2π > 0. Thisis called the straddling regime52. (b) Zoomed-in plot for neg-ative qubit-resonator detuning, the most commonly used op-erating regime.

imate) QND measurement∗∗. This limitation could belifted by implementing a pure (and not only approxi-mate) QND readout using a manifestly longitudinal cou-pling between a qubit and the resonator. Several groupsare currenly pursuing the implementation of longitudinalreadout, in which QND readout could be performed evenwith larger number of resonator photons, thus improvingthe SNR107,321,322.

We can also interpret the dispersive qubit-resonatorinteraction in another way; by rearranging the terms inEq. (144), we can equivalently write

Hdisp = ωr

(a†a+ 1

2

)+1

2

(ωq+ g2

∆︸︷︷︸Lamb shift

+ 2g2

∆ a†a︸ ︷︷ ︸ac-Stark shift

)σz,

(146)

∗∗It has been demonstrated that it is still possible to read out thequbit state by applying a very strong resonator drive tone, even-though this readout scheme is not QND320

RF LO

ADC

ADC

LORF

IF

LORF

IF

DSP

DSPLPF

LPF

FIG. 21. Schematic of an I-Q mixer. A readout pulse atfrequency ωRO enters the RF port, where it is equally splitinto two paths. A local oscillator at frequency ωLO entersthe LO port, where it is equally split into two paths, oneof which undergoes a π/2-radian phase rotation. To performanalog modulation, the two signals in each path are multipliedat a mixer, yielding the outputs I(t) and Q(t), each havingfrequencies ωRO±ωLO. I(t) andQ(t) are then low-pass-filtered(time averaged) to yield IIF(t) and QIF(t) at the intermediatefrequency ωIF = |ωRO−ωLO|, and subsequently digitized usingan analog-to-digital (ADC) converter. If ωIF 6= 0, then digitalsignals IIF[n] and QIF[n] are further digitally demodulatedusing digital signal processing (DSP) techniques to extractthe amplitude and phase of the readout signal.

where the bare qubit frequency is shifted by a fixedamount g2/∆, known as the Lamb shift†† as well as anamount proportional to the number of photons populat-ing the resonator52,214. This effect is known as the ac-Stark shift. It has the consequence that photon numberfluctuations (noise) in the resonator induce small shiftsof the qubit frequency, slightly bringing the qubit out ofits rotating frame and thus causing dephasing146. Thismeans that spurious photon occupation and fluctuationin the resonator, be it thermal or coherent photons, shiftthe qubit frequency and causing dephasing318,323. Forthis reason, it is important to make sure that the proces-sor is properly thermalized106, and its control lines wellfiltered324 and attenuated147, to reduce photon numberfluctuation.

B. Measuring the resonator amplitude and phase

In the previous section, we outlined the underlyingphysics behind the dispersive readout technique, in whichwe concluded that the qubit induces a state-dependentfrequency shift of the resonator. We now focus our at-tention on how to probe the resonator to “read out the

††It is worth mentioning that the observed qubit frequency is alwaysthe Lamb-shifted frequency and not the bare qubit frequency.

43

qubit,” that is, best distinguish the two classical res-onator signatures corresponding to our qubit states, seeFig. 19(b)-(c).

The readout circuit can be set up in measuring ei-ther reflection or transmission. The best state discrim-ination is obtained by maximizing the separation be-tween the two states in the (I,Q)-plane, i.e. the in-phase and quadrature component of the voltage, seeFig. 19(c). It can be shown that this separation ismaximal when the resonator is probed just in-betweenthe two qubit-state dependent resonance frequencies162,ωRF = (ω|0〉r + ω

|1〉r )/2. In this case, the reflected magni-

tude is identical for |0〉 and |1〉, and all information aboutthe qubit state is encoded in the phase θ, see dashedline in Fig. 19(b). In turn, the qubit-resonator detun-ing should be designed to obey the criterion for maximalstate visibility, χ = κ/2, which is maximized for phasemeasurements while constraining qubit dephasing.

Once we have picked the resonator probe frequency, thequantum dynamics of the qubit can be mapped onto thephase of the classical microwave response. In the follow-ing, we discuss how we can use heterodyne detection tomeasure the phase of the resonator response. We assumethat the reader is already somewhat familiar with basicmixer operations, such as modulation and de-modulationof signals. For interested readers, we refer to Ref. 325.

1. Representation of the readout signal

A readout event commences with a short microwavetone directed to the resonator at the resonator probe fre-quency ωRO. After interacting with the resonator, thereflected (or transmitted) microwave signal has the form

s(t) = ARO cos(ωROt+ θRO), (147)

where ωRO is the carrier frequency used to probe the res-onator. ARO and θRO are, respectively, the qubit-state-dependent amplitude and phase that we want to measure.One can equivalently use a complex analytic representa-tion of the signal,

s(t) = ReAROe

j(ωROt+θRO), (148)

= Re ARO cos(ωROt+ θRO) + j sin(ωROt+ θRO)

where Re takes the real part of an expression, e.g.,Re[exp(jx)] = Re(cosx+ j sin x) = cosx.

To gain intuition, we can rewrite Eq. (148) in a static“phasor” notation that separates out the time depen-dence exp(jωROt),

s(t) = Re

AROejθRO︸ ︷︷ ︸

phasor

ejωROt

, (149)

where the phasor ARO exp(jθRO) ≡ ARO∠θRO is a short-hand that fully specifies a harmonic signal s(t) at a known

frequency ωRO. To perform qubit readout, we want tomeasure the “in-phase” component I and a “quadrature”component Q of the complex number represented by thephasor,

AROejθRO = ARO cos θRO + jARO sin θRO (150)

≡ I + jQ (151)

to determine the amplitude ARO and the phase θRO.

2. I-Q mixing

One direct means to extract I and Q is to perform ahomodyne or heterodyne measurement using an analog I-Q mixer. Figure 21 shows a basic electrical schematicof an I-Q mixer. The readout signal s(t) and a refer-ence local-oscillator signal y(t) = ALO cosωLOt are fedinto the mixer via the RF and LO mixer ports. Themixer then equally splits the signal and local oscillatorinto two branches and multiplies them in the followingway: in the I-branch, the signal sI(t) = s(t)/2 is mul-tiplied by the local oscillator yI(t) = (ALO/2) cosωLOt;and in the Q-branch, the signal sQ(t) = s(t)/2 is multi-plied by a π/2-phase-shifted version of the local oscilla-tor, yQ(t) = −(ALO/2) sinωLOt. The “-” sign arises fromthe choice of using a A(cosωt + φ) as the reconstructedreal signal. At the I and Q ports, the output signals I(t)andQ(t) contain terms at the sum and difference frequen-cies, generally referred to as an intermediate frequency,ωIF = ωRO ± ωLO. The resulting signals are low-pass fil-tered, passing only the terms at the difference frequency,IIF (t) and QIF (t), which are then digitized. After digitalsignal processing, one obtains the static in-phase (I) andquadrature (Q) components, from which one calculatesthe amplitude ARO and the phase θRO.

Microwave mixers use square-law-type diodes to im-plement multiplication. The optical analog of a mixeroperation is a combination of a balanced (50-50) beam-splitter followed by optical photodetectors, as shown inthe inset of Fig. 22(a). The signal and local-oscillator op-tical fields are first combined at the beamsplitter, yield-ing superpositions of both fields, and then detected atthe photodetectors, which act as square-law devices. Tobuild intuition for how this works, tbe square of the sumof two electric fields (E1 +E2)2 = E2

1 +E22 + 2E1E2 has

a cross term that is the multiplication of the two fields.We refer the reader to Ref. 326 for further details.

3. Homodyne demodulation

One direct means to extract I and Q is to perform amicrowave homodyne measurement using an analog I-Qmixer of the type shown in Fig. 21. In an analog homo-dyne measurement, the local oscillator (LO) is chosen tobe at the carrier frequency ωLO = ωRO. Upon mixing, I(t)and Q(t) contain terms at both DC (ωIF = 0) and terms

44

(a)

-1

0

1

-1

0

1

LO

(a.u

.)

(a.u

.)

(b) (c)

FROMCRYOSTAT

ANALOG DEMOD.

106

DATA SAMPLING DIGITAL SIGNAL PROCESSING (DSP)

Integrate

R QIL

ωRO

ωLO

ωIFs(t)

Q

I

ωRO

ωLO

ωIF

τrd τs

Time, τ (a.u.)

FIG. 22. Schematic of the heterodyne detection technique. (a) The signal with frequency ωRF from the cryostat is mixed witha carrier tone with frequency ωLO, yielding two quadratures at a down-converted intermediate frequency ωIF = |ωRO−ωLO|, and90 out-of-phase with each other. (b) The two signals are passed into two different analog-to-digital converter (ADC) channels.To avoid sampling the resonator transient, some readout delay (τrd) corresponding to the resonator linewidth may be added,and the two signals are sampled for a time τs. In this case, the white dots represent the sampled points. (c) The sampledtraces are post-processed and after some algebra, the sampled data points are averaged into a single point in the (I,Q)-plane.To extract statistics of the readout performance, i.e. single-shot readout fidelity, a large number of (I,Q)-records are acquired,yielding a 2D-histogram, with a Gaussian distributed spread given by the noise acting on the signal.

at twice the carrier frequency. Time-averaging (filtering)I(t) and Q(t) directly yield the DC terms IIF (t) = I andQIF (t) = Q:

I = 1T

∫ T

0dt sI(t)yI(t)

= AROALO

8 cos(θRO), (152)

Q = 1T

∫ T

0dt sQ(t)yQ(t)

= AROALO

8 sin(θRO), (153)

where T is a time interval taken to be an integer num-ber of periods of the readout signal. I and Q are thensampled and used to calculate the amplitude and phase:

ARO ∝√I2 +Q2, (154)

θRO = arctan(Q/I). (155)

Note that the global value of ARO or θRO is not what mat-ters; what matters is the change in ARO and θRO betweenthe qubit being in state 0〉 and state 1〉. For example,the value of A leaving the resonator and the value G×Areaching a measurement stage are different, where G rep-resents the net gain in the measurement amplifier chain.However, the gain is the same, independent of the qubitstate, whereas A may be different, e.g., A(0)

RO = G × A|0〉

or A(1)RO = G × A|1〉. Similarly, the propagation phase

φ accumulated while a signal travels between the res-onator and the measurement stage is also independentof the qubit state, and simply imparts a phase offset tothe qubit-induced phase shift, e.g., θ(0)

RO = θ|0〉 + φ orθ

(1)RO = θ|1〉 + φ.Homodyning works in principle, but there are two

drawbacks. First, signals directly demodulated to DCmay be subject to lower signal-to-noise ratios, since theyfight against 1/f electronics noise, as well as any othernoise signals that may have inadvertently been demod-ulated (e.g., via a square-law detector). The second isthat homodyning is not compatible with frequency di-vision multiplexing (FDM), where a single pulse can beused to interrogate N resonators at different frequenciesby applying tones at each resonator frequency using thesuperposition principle, e.g.,

s(t) =N∑i=1

A(i)RO cos(ω(i)

ROt+ θ(i)). (156)

Homodyning an FDM signal will put all resonator signalsat DC, and once downconverted, they cannot be differen-tiated. To work around this, it is generally advantageousto use heterodyning, which uses a two-step demodulationprocess via an intermediate frequency ωIF. Such a schemeis easily compatible with the concept of FDM, because areadout signal is first demodulated to unique IF frequen-

45

cies ω(i)IF , and then digitally demodulated to extract each

A(i)RO and θ(i). In the following, we will consider N = 1

for simplicity, but the process is applicable to larger Nprovided the frequencies a sufficiently spaced to avoidinterference with one another during the demodulationprocess.

4. Heterodyne demodulation

In a heterodyne scheme, a local oscillator at frequencyωLO is offset by an intermediate frequency ωIF to target aunique readout frequency ωRO. Up-conversion techniquessuch as single-sideband modulation with suppressed car-rier (SSB-SC) using balanced I-Q mixers (operated in re-verse compared with Fig. 21) are commonly used to cre-ate such readout signals. We refer the reader to Ref. 325for more information on how to create such pulses.

Here, we want to extract ARO and θRO (or their scaledand offset versions) from the reflected/transmitted toneusing a heterodyning scheme. The first step is to per-form analog I-Q mixing, as illustrated in Fig. 22(a). Incontrast to the homodyning case, here, the local os-cillator and readout tone are at different frequencies,ωIF = |ωRO − ωLO| > 0. Mixing the LO and RO sig-nals yields the signals I(t) and Q(t) with terms at bothsum and difference frequencies. Filtering out the sum fre-quencies using low-pass filtering (time averaging) yieldsthe IF signals:

IIF(t) = 1T

∫ T

0(dt) sI(t)yI(t)

= AROALO

8 cos(ωIFt+ θRO) (157)

QIF(t) = 1T

∫ T

0(dt) sQ(t)yQ(t)

= AROALO

8 sin(ωIFt+ θRO). (158)

As before, we have omitted any offset phases from theLO or from the wave propagation between the resonatorand the measurement. Again, these offset values are notwhat matters; it is the change in ARO and θRO with achange in qubit state that allows state discrimination.

The analog-demodulated IIF(t) and QIF(t) are now os-cillating at a frequency that is generally low enough tobe digitized using commonly available analog-to-digitalconverters (ADCs). The resulting digital signals are nowwritten as IIF[n] and QIF[n],

IIF[n] = AROALO

8 cos(ΩIFn+ θRO) (159)

QIF[n] = AROALO

8 sin(ΩIFn+ θRO), (160)

where n = t/∆t indexes the sample number of thecontinuous-time signals IIF(t) and QIF(t), ΩIF = ωIF∆t isthe digital frequency, and ∆t is the sampling period (typ-ically around 1 ns). Pulsing the resonator is necessarily

accompanied by a ring-up time, related to the qualityfactor of the resonator, and the first few samples maydecrease overall signal-to-noise. Consequently, a delayedwindow of samples [n1 : n2] is often used to perform thesecond digital demodulation of the discrete-time signalsIIF[n1 : n2] and QIF[n1 : n2]. Note that more compli-cated windowing functions may also be used to improvestate discrimination, but here we use a simple boxcar [seeFig. 22(b)].

Digital demodulation comprises the point-by-pointmultiplication of IIF[n1 : n2] and QIF[n1 : n2] by cos ΩIFnand sin ΩIFn. Averaging the resulting time series elimi-nates the 2ΩIF component while retaining the DC com-ponent, as in a homodyne measurement, one obtains

I = 1M

n2∑n1

IIF[n] cos[ΩIFn] = AROALO

16 cos θRO, (161)

Q = 1M

n2∑n1

QIF[n] sin[ΩIFn] = AROALO

16 sin θRO, (162)

where M = n2 − n1 + 1. As before, I and Q can then beused to find ARO and θRO.

The same procedure may be view in the complex I−Qplane by the analytic function zIF[n], as illustrated inFig. 22(c-d),

zIF[n] = IIF[n] + jQIF[n] ≡ VI [n] + jVQ[n] (163)

= AROALO

8 [cos(ΩIFn+ θRO) + j sin(ΩIFn+ θRO)](164)

= AROALO

8 ejθROejΩIFn (165)

where the digital in-phase and quadrature signals are rep-resented here as the voltages VI [n] and VQ[n] sampledby the ADC, and we have separated the static phasor(AROALO/8) exp[jθRO] from the rotating term exp[jΩIFn].One can digitally demodulate the time series zIF[n] bymultiplying by the complex conjugate of the oscillatoryexponential,

z[n] = zIF[n]. ∗ e−jΩIFn (166)

where .∗ indicates a point-by-point multiplication, andthe result is a vector of length M of nominally identicalvalues of the phasor – one for each sample point – with asmall amount of additive noise due to noise in measure-ment chain, digitization errors, etc. A singular phasorvalue is then estimated by taking average,

z[n] = 1M

∑z[n] (167)

= AROALO

8 ejθRO . (168)

Such “single-shot measurements” may then be repeateda large number of times to obtain an ensemble average〈z[n]〉.

46

C. Weak and strong qubit measurements: Impact of noise

In quantum measurements, noise plays an essential roleas it dictates the fidelity of its outcome127,327, recall Fig.22(c). In the absence of noise, any non-zero dispersiveshift (resulting in a resonator field displacement) wouldsuffice to unambigously separate the qubit states, givena properly chosen resonator linewidth. In practice, how-ever, the outcome of the quantum measurement is gener-ally Gaussian distributed in the (I,Q)-plane due to pres-ence of classical and quantum noise. In this section, wereview the main sources of noise, as well as how it im-pedes our ability to extract information from the quan-tum system. For a rigorous discussion of noise and quan-tum measurements, the interested reader is referred e.g.to the work by Clerk et al127 and to the textbook byHaus126.

The total noise added to the signal has multiple origins.One part of the noise is associated with the microwavesignal used to probe the resonator, where each photonhas an intrinsic quantum noise power of ~ω/2 per unitbandwidth. Another contribution comes from the phase-preserving amplifiers, adding both classical noise and atleast ~ω/2 of noise as required by Heisenberg’s uncer-tainty relation. Finally, any attenuation of the signalprior to the first amplifier will appear as added noise.Combined, these noise sources amount to a system noisetemperature, which can be characterized using a sensitivethermometer, such as a shot-noise tunnel junction328 ora qubit329 as a sensor.

The noise results in time-dependent fluctuations of themeasured signal, which in turn translates into uncer-tainty in our demodulated signals, see Fig. 22(c). Thiscan be intuitively understood by considering that ourheterodyne detection method requires us to sample for afinite amount of time.

To quantify the impact of the noise on our measure-ment, we first project the distributed (I,Q) data – cor-responding to |0〉 and |1〉 – onto the axis for which theirrelative separation in the complex plane is maximized‡‡.The line that is used to separate between |0〉 and |1〉 iscalled a separatrix.

The noise can now be quantified by comparing thewidths of the Gaussian probability distribution sur-rounding the mean with the peak separation in the(I,Q)-plane, thus defining a signal-to-noise ratio SNR =δθ/(∆θ1 +∆θ0), see Fig. 23(a), with δθ = |θ1−θ0| repre-senting the signal and ∆θ0, ∆θ1 represent the noise (2σ)of each distribution. The SNR allows us to distinguishbetween a weak and a strong quantum measurement, asillustrated in Fig. 23(b)-(d).

In a weak measurement, the probabilities are broadlydistributed as compared to their relative separation

‡‡When analyzing the readout data, we have the freedom to chooseprojections.

(a)

(b) (c)

Weak: SNR < 1

Strong:SNR > 1

Counts (a.u.)

Sampling time, τs (a.u.)

θ1

θ0

Inte

grat

ed d

et. s

igna

l (a.

u.)

Inte

grat

ed d

et. s

igna

l (a.

u.)

Counts (a.u.) Counts (a.u.)

(d)

FIG. 23. (a) Qubit state distribution throughout the courseof sampling the readout signal, in the presence of noise. Theseparation between the peaks (solid lines) increases linearly intime, whereas the peak widths only increase as

√t. Image in-

spired by Clerk et al127. The three black arrows represent linecuts for three sampling times: (b) For short sampling time,the states are not separated, resulting in a weak meaurement(SNR < 1). (c) After a longer sampling time, the peaks startsto get separated, (d) finally getting fully resolved, resultingin a strong measurement (SNR > 1).

(SNR < 1), which means that only partial informationof the quantum state is revealed to the observer, seeFig.23(b). In a strong measurement, on the other hand,the quantum state is collapsed onto one of the two eigen-states. In this case, the outcome of the measurement canbe distinguished unambigously, which is reflected in twofully separated distributions (SNR > 1), see Fig. 23(d).

In many applications of quantum measurements, itis necessary to unambigously (and with high fidelity)tell the outcome without repeating the readout measure-ment. This is known as single-shot readout and it oftenrequires the use of a parametric amplifier – a preamplifierused to increase system SNR – which is further discussedin Sec. V E 3.

Assuming that the widths of the two distributions areidentical, ∆θ0 = ∆θ1 = ∆θ, the separation error canbe calculated by deriving the weight of the overlappingregion of the Gaussian distributions as162

47

εsep = 1√2π(∆θ)2

∫ ∞θ= θ0−θ1

2

exp[− (θ − θ1)2

2(∆θ)2

]dθ

= 12erfc

[|θ0 − θ1|

2√

2(∆θ)2

], (169)

where erfc(x) denotes the complementary Gaussian errorfunction, defined as

erfc(x) = 1− 2√π

∫ ∞x

e−t2dt. (170)

Using the erfc in Eq. (170), the separation error inEq. (169) can be compactly expressed in terms of thesignal-to-noise ratio,

εsep = 12erfc

[SNR

2

](171)

Note, however, that the separation error between thetwo state distributions only tells us the signal-to-noiseratio of our detection scheme. On top of the separationerror, fidelity is reduced if the qubit relaxes (or is excited)during the readout. This will result in a count on the“wrong” side of the threshold. This leads to an additionalconstraint on the readout; The readout cycle needs to becompleted on a timescale much shorter than the qubitrelaxation time.

In summary, we see that to optimize the qubit read-out fidelity, the readout needs to fulfill the following tworequirements:

• Fast readout: The readout cycle needs to be com-pleted within a time that is short compared withthe qubit coherence time. The longer the readouttime, the more likely the qubit is to relax, thus re-ducing readout fidelity.

• High signal-to-noise ratio: The signal-to-noiseratio needs to be sufficiently large to suppress thestate separation errors below an acceptable limitwhere it does not limit the readout fidelity.

In sections V D and V E 3, we review how these twoconditions are met by carefully engineering the signalpath of the readout circuitry.

D. “Purcell filters” for faster readout

To ensure high-fidelity readout performance, it is im-portant to perform single-shot readout at a timescalemuch shorter than the qubit coherence time, τro T1.This motivates us to: (i) make the resonator linewidthwide, thus reducing its ring-up time, τrd, and (ii) keep

the integration time τs as short as possible, see Fig.22(b).The ability to isolate a quantum system from decoheringinto its environment while, at the same time, being ableto read out its state in a short time represents two con-tradictory criteria, which must be traded-off327.

Even though dispersive readout (in the few-photonlimit) has only a small back-action on the qubit state,the qubit will still suffer from T1-relaxation while we areperforming a measurement. In fact, this “decay duringthe readout” often limits the readout fidelity, reducing itto

F (τro) = 1− e−τro/T1 , (172)

where τro = τrd + τs/2 denotes the total time for thereadout, consisting of the readout delay τrd due to theresonator transient, and half the sampling time τs/2. Thefidelity drop in Eq. (172) can be interpreted as a man-ifestation of the competition between the time scales atwhich our quantum information reaches our detector orthe environment first.

The limitation of qubit coherence originates from anenhanced spontaneous emission of photons, induced byits environment. This is known as the Purcell effect330,and is an important consideration when designing qubit-resonator systems331. The portion of spontaneous emis-sion that is mediated by the resonator describes howqubit relaxation is enhanced by the resonator Q when on-resonance, and suppressed off-resonance. The aim of thissection is twofold: first, we develop an intuition for howthe Purcell decay limits qubit coherence, and second, howto properly mitigate this limitation by designing a so-called Purcell filter, which modifies the impedance seenby the qubit through the readout resonator. This allowsus to maintain fast readout, while protecting the qubitfrom relaxing into its environment.

If we would just choose qubit and resonator operationfrequencies guided by the resonator linewidth κ, qubit-resonator coupling g, and the amount of dispersive shiftχ, we would reduce the detuning between the qubit andthe resonator, thus maximizing the dispersive shift (recallFig. 20). However, this presents a trade-off between twoimportant system parameters; on one hand, we want thequbit to be isolated from the resonator environment off-resonant to avoid Purcell-enhanced decay. On the otherhand, looking at the dispersive shift, we want the tworates, g and κ to be strong, yielding larger dispersiveshift as well as short resonator transient and thus a fasterreadout.

Fortunately, when operating in the dispersive regime,the qubit and resonator are far detuned from each other∆ g, κ, which means that their impedance (environ-ment) can be independently engineered through filterdesign. In essence, one designs a filter to have strongcoupling to the readout port at the resonator frequency(large κ), but isolates the qubit from its environment atthe qubit frequency332,333. In other words, an impedancetransformation.

48

Depending on the design of the readout for the quan-tum processor to which the filter should be coupled,there are different ways to design a Purcell filter; such asquarter-wave stubs332, low-Q bandpass filters66,333, andstepped-impedance filters334. Which one is optimal de-pends on system properties such as qubit-resonator de-tunings, required bandwidth, and allowed insertion loss.

The most promising Purcell filter designs are the onesthat allow for frequency multiplexing, such as the low-Qbandpass filter design66,333, which in addition to Purcellfiltering has the function of a quantum bus, connectingseveral frequency-multiplexed readout resonators sharingthe same amplifier chain.

The Purcell effect can be framed in terms of Fermi’sgolden rule, where noise in the environment causes thequbit to decay with some probability. We can gain intu-ition about the Purcell effect (as well as how the qubit canbe protected from it) by replacing the Josephson junctionin the qubit circuit with an ac-current source, outputtingI(t) = I0 sin(ωt), with I0 = eω and study the rate atwhich power is lost into an environmental load resistorR = Z0 = 50 Ω, see Fig. 24(a).

Expressing the power lost in the resistor as P =I20 (Cg/CΣ)2R = (eωβ)2Z0, with β = Cg/CΣ, the qubit

Purcell decay rate into the continuum can be written as

γPurcellenv = 1

T1= P

~ω= (βeω)2Z0

~ω= g2

ω. (173)

To protect the qubit from decaying into the 50 Ω envi-ronment (as well as for deploying our dispersive readout)we can now add a resonator in parallel with the qubit,see Fig. 24(b). The presence of the resonator has theeffect of shaping the impedance at the qubit frequency,which in turn modifies the decay rate in Eq. (173) into

γPurcellres-env = g2

ω

Re[Zr(ω)]Z0

, (174)

where Zr(ω) denotes the impedance of the shunted res-onator. We can express the real-part of the impedancein terms of the resonator quality factor Q = ωr/κ andqubit-resonator detuning ∆ = ωq − ωr,

Re[Zr(ω)] = QZ0

1 + 2(∆/κ)2 . (175)

Now, by substituting Eq. (175) into Eq. (174), wesee that the Purcell decay rate for the qubit depends onthe detuning between the resonator and the qubit. Thisis intuitive, since the resonator can be thought of as abandpass filter, with center frequency ωr and bandwidthκ. For resonant condition, i.e. when ∆ = 0, the emissionrate into the resonator takes the form

γPurcellres-env = g2

ωr

Re[Zr]Z0

=∆=0

g2

ωrQ = g2

κ. (176)

In the dispersive regime ∆ g, κ, which is also rele-vant for us in the context of qubit readout, we can makethe approximation Re[Zr] ≈ QZ0(κ/∆)2, yielding thefamiliar expression for the Purcell decay rate in circuitQED331

γPurcellres-env = g2

ωr

Re[Zr]Z0

=∆g,κ

g2

ωrQ( κ

∆

)2=( g

∆

)2κ.

(177)The relation for the Purcell limit in Eq. (177) thus

provides us with a useful guide on how to design the cou-pling rates g and κ, as well as how large qubit-resonatordetuning ∆ is necessary to avoid the Purcell limit.

In recent years, however, the intrinsic coherence timesfor superconducting qubits have reached above 100µs,recall Sec. II, imposing practical limitations on how tosimultaneously optimize g and κ, to render fast readoutwithout compromising the qubit coherence. Consideringthe parameters in Eq. (177), it is not possible to justincrease the bound on the relaxation time T1, without atthe same time trading off the readout speed and contrast.

We can now introduce the Purcell filter [Fig. 24(c-d)]in between the readout resonator and the 50 Ω environ-ment, leading to a reduction of the decay rate accordingto332

γPurcellres-filter-env = κ

(g

∆

)2(ωq

ωr

)(ωr

2QF∆

), (178)

where QF denotes the quality factor of the Purcell filter.This is schematically depicted in Fig. 24(d), where thePurcell filter is placed around the resonator frequency,while far detuned from the qubit.

E. Improve signal-to-noise ratio: Parametric amplification

In light of the aforementioned limited signal-to-noiseratio associated with the low photon number of the dis-persive qubit readout, and the short sampling time, thenoise temperature of the amplifier chain plays a crucialrole in determining the fidelity of the measurement.

A useful benchmark for quantum measurements is thequantum efficiency, defined as

ηSQL = ~ωRF

kBTsys, 0 < ηSQL < 1, (179)

which quantifies the photon energy to the system noisetemperature Tsys, thus yielding a measure of how closethe signal is to the standard quantum limit (SQL), asimposed by Heisenberg’s uncertainty relation, adding1/2 photon of noise when ηSQL approaches unity. Sincethe energy of each microwave photon is much smallerthan that of optical photons, it is not easy to builda single-photon detector operating in the microwave

49

(a)

Qubit Qubit

(b)

(c)

Env. Env.Res.

Qubit Env.Res. Purcell filter

(d)

Mag

nitu

de (a

.u.)

Frequency, (a.u.)

Purcell filter

ResonatorQubit

g κ

g κ

CgZ0CΣ

I(t) I(t)g

I(t)

CΣ

ωqωq

ωq ωr ωp

Z0

Z0

ωr

ωq ωr

FIG. 24. (a) Circuit representation of qubit (orange) cou-pled to an environment (blue) with a load resistor, Z0, via acapacitor Cg. To study the decay rate, the Josephson junc-tion has been replaced with a current source, I(t). (b) Byadding a resonator (red) with frequency ωr in-between thequbit and the 50 Ω environment, we get the case found in reg-ular dispersive readout. (c) A Purcell-filter (green) is addedto the circuit, providing protection for the qubit, while al-lowing the resonator field to decay fast to the environment.(d) Transmission spectrum of a Purcell filter (dashed green),centered around the resonator frequency (red arrow), whereasthe qubit frequency (orange arrow) is far detuned.

domain335,336. Instead, for heterodyne detection in cir-cuit QED, a set of cascaded microwave amplifiers areused. The system noise temperature for the amplifierchain can be expressed in terms of the individual gain fig-ures Gn and noise temperatures TN,n of each constituentamplifier337

Tsys = TN,1 + TN,2G1

+ TN,3G1G2

+ ... (180)

where n = 1, 2, 3, ... denotes the order of the amplifiers,starting from the qubit chip. From Eq. (180), we see thatthe noise temperature Tsys is dominated by the noise con-tribution from the first amplifier, whereas the gain of thefirst amplifier has the effect of suppressing the noise fromthe second amplifier, and so on. If the first amplifier is alow-noise high-electron mobility transistor (HEMT) am-

plifier (TN ≈ 2 K), the system noise temperature whenimplemented in a cryostat is around 7-10 K, correspond-ing to around 10-20 added photons of noise per signalphoton around 5 GHz. In practice, this is generally toomuch noise to perform single-shot readout.

This inherently poor signal-to-noise ratio has revivedinterest in developing quantum-limited parametric am-plifiers (PA) – tailored for readout of superconductingqubits – featuring the ability to amplify small microwavesignals, and adding only approximately the minimumamount of noise allowed by quantum mechanics126,127,327.

1. Quantum-limited amplification processes

In a linear, phase-insensitive amplifier, an input state〈ain〉 is amplified to an output state 〈aout〉, with an am-plitude gain factor

√G. Microwaves are electromagnetic

fields and therefore considered to be coherent light com-prising microwave photons. As such, they must obey thecommutation relations126,327,338,339

[ain, a†in] = [aout, a

†out] = 1, (181)

from which it can be shown that it is not possible tosimultaneously amplify both quadratures of ain withoutalso adding noise. This is known as Caves theorem afterthe work by Caves327, based on earlier work by Haus andMullen338. This can be seen by considering the scatteringrelation between the input and output microwave fields

aout =√Gain. (182)

The gain relation in Eq. (182) constitutes our idealscenario for an amplifier process. However, the problemis that that this relation does not satisfy the commutationrelation in Eq. (181). To satisfy this relation, we need toalso take into account the vacuum fluctuations of anothermode127,340–342 – called the idler mode bin, also satisfyingthe same communtation relation [bin, b

†in] = 1. To satisfy

the commutation relation, the idler mode is amplified bythe gain factor

√G− 1. For large gain, it can be shown

that a minimum amount of half a photon of noise ~ω/2needs to be added to a signal amplified with gain

√G.

Finally, taking the idler mode into account, the scatter-ing relation for the coherent output field takes the form

aout =√Gain︸ ︷︷ ︸

Amplification

+√G− 1b†in︸ ︷︷ ︸

Added idler noise

. (183)

Generally, this process results in a so-called phase-insensitive parametric amplification process, in whichboth quadratures of the input field gets equally ampli-fied. This is illustrated in Fig. 25, where the in-phase(Iin) and quadrature (Qin) components of the fields areplotted, before and after the parametric amplifier.

50

AMPLIFIER INPUT AMPLIFIER OUTPUTMIXING PROCESSES

Pump

Signal

Idler

Vac. noise

(a) (b) (c)

vac.

ωS

ωi

ωp

FIG. 25. Schematic illustration of a quantum-limited, phase-preserving parametric amplification process of a coherent inputstate, ain = Iin + iQin. (a) The state is centered at (〈Iin〉, 〈Qin〉) and has a noise represented by the radii of the circles along thereal and imaginary axes, respectively. (b) Scattering representation of parametric mixing, where the signal and pump photonsare interacting via a purely dispersive nonlinear medium. (c) In the case of phase-preserving amplification, both quadraturesget amplified by a factor

√G, while (in the ideal case) half a photon of noise gets added to the output distribution (blue).

Image inspired by Flurin339.

Considering the amplification process in Eq. (183), wecan find a special case for the idler mode, for which noise-less amplification can be accomplished for one of the twoquadratures, but at the expense of adding more noiseto the other, thus not violating Heisenberg’s uncertaintyrelation for the two field quadratures. This mode of op-eration is known as phase-sensitive amplification, and isobtained when the idler mode oscillates at the same fre-quency as the signal (or a multiple thereof), but can beshifted with an overall phase φ ∈ [0, 2π]. By substitut-ing the idler mode in Eq. (183) with bin = eiφain, thescattering relation becomes

aout =√Gain︸ ︷︷ ︸

Amplification

+ e−iφ√G− 1a†in︸ ︷︷ ︸

Phase-dep. noise

. (184)

The overall phase factor allows us to tune the orienta-tion of the amplification (or de-amplification) by meansof the pump phase, thus allowing us to choose a quadra-ture for which we want to reduce the noise, see Fig. 26.Intuitively, this can be understood by considering the in-terference that occurs when two waves with the same fre-quency are confined in space, where we obtain construc-tive or destructive interference, depending on the phasebetween the two waves. Due to this interference, thenoise can be suppressed even below the standard quan-tum limit (without violating Heisenberg’s uncertainty re-lation). This is known as single-mode squeezing and wasfirst observation in superconducting circuits by Yurke etal.343. In particular, after the theoretical prediction byGardiner344, Murch et al. showed that the coherencetime of a qubit can be enhanced when the qubit is ex-posed to squeezed vacuum345,346. Also two-mode squeez-

ing was demonstrated by Eichler et al.347, where the de-modulation setup squeezes both quadratures of the ac-quired signal107.

In the context of qubit readout, however, phase-sensitive amplification tends to be experimentally in-convenient. This is mainly due to its phase-dependentgain, which imposes stringent requirements on continu-ous phase-calibration of the readout signal.

For a detailed theoretical framework developed forquantum limited amplification, the reader is referred toearlier work by Roy and Devoret348, Clerk et al.127, andWustmann and Shumeiko349.

2. Operation of Josephson parametric amplifiers

In this section, we review the basic operation charac-teristics of parametric amplifiers, and in particular theJosephson parametric amplifiers (JPAs), that have beenexploited for qubit readout. Although many differentflavors of parametric systems exist, we here focus on theresonant implementations of the Josephson parametricamplifier (JPA), serving as a good system for reviewingthe fundamental concepts around parametric amplifica-tion.

All parametric amplifiers operate based on one funda-mental principle: the incoming signal photons are mixedwith an applied pump tone via an intrinsic nonlinearity,by which energy from the pump is converted into signalphotons and thereby providing gain. As we recall fromSec. II, such a nonlinearity can be engineered in the mi-crowave domain using Josephson junctions350, and theresonant parametric amplifiers are built from slightly an-harmonic oscillators.

51

(a)

3 /2

(b)

10

20

30

0

-10

-20

-30

Pump-phase angle, θ (rad)5 /2 7 /2 9 /2 /2

Volta

ge g

ain,

|G| (

dB)

FIG. 26. Phase-sensitive parametric amplification. (a) Incontrast to the phase-insensitive operation, phase-sensitiveparametric amplification allows us to suppress the noise alongone axis. Consequently, the noise is added to the otherquadrature. (b) Voltage gain as a function of pump-phaseangle, in which the amplification depends on the phase of thepump, providing either amplification or de-amplification ofthe quadrature voltage.

The first Josephson parametric amplifiers were builtfrom a coplanar waveguide resonator, made nonlinear byadding a nonlinear Josephson contribution to its totalinductance, see Fig. 27(a). The word parametric refersto the process of modulating (or pumping) one of theparameters of the system’s equation-of-motion (such asfrequency or damping) in time349,351,352. The naturalway to perform this parametric pumping is to modulatethe nonlinear Josephson inductance, which in turn hasthe effect of modulating the resonator frequency ωr(t) =1/√L(t)C.

Depending on how the pumping is implemented, thereare two different mixing processes that can be exploitedin Josephson parametric amplifiers, which determinesthe characteristics of the amplifier. These are illus-trated in Fig. 27(b)-(c) and are referred to as current-pumping351,353–356 and flux-pumping95,349,352,357–362, re-spectively. The type of mixing process that takes placedepends on the leading order of the nonlinearity of thesystem, as reflected in its Hamiltonian. In the follow-ing, we briefly review the difference between these twopump-schemes.

In the current-pumped case, the dynamics of the sys-tem has characteristics of a Duffing oscillator363, with afourth-order nonlinear term in addition to the harmonicoscillator term in its Hamiltonian

Resonator

(a) Josephson parametric amplifier (JPA)

SQUID

Current-pumping (4-wave mixing): 2ωp = ωs + ωi (b)

Flux-pumping (3-wave mixing): ωp = ωs + ωi (c)

Inp

ut

+ P

um

pP

um

pIn

pu

t

ωs ωr 2ωr ωs ωr 2ωrωi

ωs ωr ωiωs ωr

κ

|ain|

ωp G|ain|ωp

G|ain||ain|

ωp

ωp

FIG. 27. Circuit schematics and pump schemes of a Joseph-son parametric amplifier. (a) The device consists of a quarter-wavelength resonator (blue), represented as lumped elements,shorted to ground via a Kerr-nonlinearity consisting of twoparallel Josephson junctions (orange) forming a SQUID. Thepump (red) can be applied in two ways; (b) either by mod-ulating the current through the junctions (four-wave mixing)at the resonant frequency, ωp ≈ ωr, or (c) by modulating theac-flux Φac around a static dc-flux point Φdc using a separatefast-flux line (three-wave mixing). The flux pump is appliedat twice the resonant frequency, ωp ≈ 2ωr.

H = ωrc†c+Kc†c†cc, (185)

where c denotes the resonator field operator and K is the“Kerr-nonlinearity”. This process is a so-called four-wavemixing process, since it mixes four photons: one signal(ωs), one idler (ωi), and two pump photons (ωp), obeyingthe energy conservation relation ωs + ωi = 2ωp, see Fig.27(b). Pioneered by Yurke351, this was the first demon-stration of microwave amplification using a Josephsonparametric amplifier. When the signal and idler modesare at the same frequency, the amplification is said to bedegenerate. This pumping scheme is the foundation forthe Josephson Bifurcation Amplifier (JBA), developedby Siddiqi et al.355,364,365, which has been used to per-form single-shot qubit readout, by mapping the quantumstates onto the high and low resonator field originatingfrom the sharp bifurcation point of the amplifier366.

In the other case, when the system is flux-pumped,the parametric process is driven by threading a magnetic

52

flux Φac through a SQUID loop, thereby modulating thefrequency of the resonator. This results in a three-wavemixing process, comprising three photons: one signal,one idler, and one pump photon, with ωs + ωi = ωp, seeFig. 27(c). Therefore, we see that the pump frequency isabout twice that of the signal ωp ≈ 2ωs for ωs ≈ ωi. Fordegenerate, flux-pumped systems, the leading nonlinear-ity is a third-order term, yielding a Hamiltonian

H = ωrc†c+K

(pc†c† + p†cc

), (186)

where the p operator denotes the flux-pump mode. Thisapproach to building parametric amplifiers was devel-oped by Yamamoto et al.358, as well as by Sandberg etal.95.

The flux-pumping scheme has several practical advan-tages. First, the large detuning of the pump makes it eas-ier to filter, isolating the readout signal as its passing intothe digitizer downstream and preventing the saturationof following amplifier stages. Second, if the resonator isa quarter-wavelength resonator, it has no resonant modeat the pump frequency ωp, reducing spurious populationor saturation of the system as well as backaction on thequbits in the processor. Third, since the flux pump lineis a separate on-chip microwave line, no additional direc-tional coupler is needed.

Due to its rich dynamics, flux-pumping has alsoproven a useful platform to study the quantum dy-namics of Josephson parametric oscillators, both in thecontext of qubit readout367–370, the dynamical Casimireffect371–373, and to better understand their complexnonlinear dynamics360,363,374–378.

In addition to the degenerate parametric interactionsdescribed above, parametric gain can be obtained be-tween different resonant modes; either between differentmodes of the same resonator328,379, or in-between differ-ent resonators380, as with the Josephson parametric con-verter (JPC)339,381–385. In addition to the possibility ofisolating and amplifying certain frequencies, the JPC canimplement frequency conversion for which it has someother areas of applications compared with other types ofparametric amplifiers.

3. The traveling wave parametric amplifier

In the previously described JPA, parametric ampli-fication is realized using resonators that enhance theparametric interaction between the input signal and theJosephson junction nonlinearity. Essentially, the Q-enhancement of the resonator forces each photon to passthrough the junction on average Q times before leavingthe resonator, thereby enhancing the non-linear interac-tion. Albeit proven to be able to reach near the standard-quantum limit of noise for readout of a small number ofqubits, the future direction of the community is headingtowards amplifier technologies which are compatible withmultiplexed readout of several qubits coupled to the same

amplifier chain65,66,386–388. In this context, resonator-based parametric amplifiers suffer from two major draw-backs: First, the amplifier bandwidth is limited to theresonator linewidth, typically ≈ 10−50 MHz, practicallylimiting the number of multiplexed frequencies that canbe amplified. Second, since the Josephson nonlinearityis realized by a small number of junctions, the satura-tion power is low due to the interplay of higher ordernonlinearities, effectively taking the system outside itsdesired operation regime363,374,375,385. In practice, thislimits how many readout resonators that can be simul-tanously read out.

These two bottlenecks can, to a degree, be overcomewith microwave engineering. For instance, the linewidthcan be made an order of magnitude wider by altering theimpedance along the resonator. This is called a stepped-impedance transformer, where the impedance is rampeddown from a matched 50 Ω at the capacitor down to asmall impedance at the SQUID389 shorting the device toground. Also the saturation power can be increased bydistributing the nonlinearity across an array consisting ofmany identical junctions, reducing the Kerr-nonlinearityby a factor 1/N2 with N representing the number of junc-tions in the array. This has been demonstrated by usinga an array of SQUIDs in a resonator, rather than a singleone359.

However, despite the above mentioned engineering ef-forts to improve the resonator-based JPAs, the mostprominent approach to date is to get rid of the resonatoraltogether and, instead, construct a microwave analog tooptical parametric amplifiers, where kilometers of weaklynonlinear fibers are used. Such device is called a travelingwave parametric amplifier (TWPA) and was developedto surmount the bandwidth and dynamic range limita-tions of the resonator-based JPAs.

Although operated in similar way, the nonlinearity ofTWPAs can be realized in different ways, such as the ki-netic inductance of a superconducting film390–392 or usingan array of Josephson junctions329,393,394, through whichthe four-wave mixing process is distributed across a non-linear lumped element transmission line, see Fig. 28(a).

The Josephson TWPA consists of a few thousand iden-tical unit cells, each comprising a shunt capacitor toground and a nonlinear Josephson inductor, togetheryielding a characteristic impedance of Z0 =

√LJ/C ≈

50 Ω, see Fig. 28(a).The fact that the nonlinearity is distributed allows for

high saturation power, since each Josephson junction isaccessed once. However, even though energy conserva-tion is satisfied, the four-wave mixing process in the de-vice, there is a problem with phase (or momentum) con-servation. This is associated with the system nonlinear-ity as well as the large frequency detuning between signaland pump photons, yielding a difference in phase-velocitybetween the two, which in turn gives rise to a non-flatgain profile, as well as an overall reduction in gain393.

Again, by taking inspiration from the dispersive en-gineering developed in quantum optics and photonics,

53

(c)

(b)

0

5

10

15

20

Gain

(dB)

0 2 4 6 8 100.00

0.05

0.10

0.15

0.20

Frequency (GHz)

Wav

evec

tor,

k (a

-1)

6.106.000.08

0.10

5.85

Ip=0.7I0Ip=0.5I0

Ip=0.3I0

(d)

(a) Josephson traveling wave parametric amplifier (JTWPA)

RPM Res.

IN OUT

RPMTWPA

TWPA

RPMTWPA

TWPA

ωs

ωp

LJC0

CC

CrLr

Z0

0 2 4 6 8 10Frequency (GHz)

FIG. 28. (a) Simplified circuit representation of a Joseph-son traveling wave parametric amplifier (JTWPA). The char-acteristic impedance for each unit cell is set by the in-lineJosephson inductor, LJ (orange) and the shunt capacitor, C(blue). A resonant LC-circuit (red) is used to phase matchthe four-wave amplification process. (b) Schematic of how thesignal gets amplified in each unit cell as it propagates throughthe device. (c) Gain vs. frequency for a JTWPA, with andwithout the resonant phase matching (RPM). (d) Dispersionrelation of the TWPA, where the LC-resonators collectivelyopen up a stopband at the resonant frequency. By applyingthe pump close to this frequency, the wave vector of the pumpcan be set to obtain a phase-matching. The optimal pumpfrequency depends on the pump power, as indicated in theinset. Image courtesy of Kevin O’Brien329,393

where the refractive index can be periodically altered toengineer the momentum of a transferred signal, the so-lution to this phase-mismatch problem was introducedby O’Brien et al.393. By introducing resonators at peri-odic intervals of TWPA unit cells, the pump tone can begiven a “momentum kick”, effectively slowing it down andphase-matching the device by means of its wave vector.

This technique is called resonant-phase matching (RPM),see Fig. 28(d), and requires that the pump frequencyis set on the left side of the dispersion feature (wherethe wave vector diverges), defined by the resonant fre-quency of the phase-matching resonators. Note, finally,that broadband parametric amplification with high dy-namic range has been demonstrated in other Josephson-based circuits, e.g. the superconducting nonlinear asym-metric inductive element (SNAIL) parametric amplifier(SPA)395.

VI. SUMMARY AND OUTLOOK

In this review, we have discussed the phenomenalprogress over the last decade in the engineering of su-perconducting devices, the development of high-fidelitygate-operations, and quantum non-demolition measure-ments with high signal-to-noise ratio. Putting these ad-vances together, we hope that it is clear that the planarsuperconducting qubit modality is a promising platformfor realizing near-term medium scale quantum proces-sors. While we have focused on highlighting the advancesmade within the fields of realizing, controlling and read-ing out planar superconducting qubits specifically usedfor quantum information processing, there has of coursealso been tremendous activity in the surrounding fields.In this final section, we briefly mention a few of thosefields, and invite the reader to look into the references,for further details.

Quantum annealing: Superconducting qubitsalso form the basis for certain quantum annealingplatforms396,397. Quantum annealing operates by find-ing the ground state of a given Hamiltonian (typicallya classical Ising Hamiltonian), and this state willcorrespond to the solution of an optimization problem.By utilizing a flux-qubit type design (see Sec.II, thecompany D-Wave have demonstrated quantum anneal-ing processors86 which have now reached beyond 2000qubits398. The benchmarking of quantum annealersand attempts to demonstrate a quantum speedup for ageneral class of problems is a highly active research field,and we refer the reader, for example, to recent papersRefs. 399–401 and references therein.

Cavity based QIP: A parallel effort to the planar su-perconducting qubits discussed in this review is the de-velopment of 3D cavity-based superconducting qubits. Inthese systems, quantum information is encoded in super-positions of coherent photonic modes of the cavity99. Thecat states can be highly coherent due to the inherentlyhigh quality factors associated with 3D cavities100,402,403.This approach has a fairly small hardware overhead to en-code a logical qubit404, and lends itself to certain imple-mentations of asymmetric error-correcting codes due tothe fact that errors due to single-photon loss in the cavityis a tractable observable to decode. Using this architec-ture, several important advances were recently demon-strated including extending the lifetime of an error-

54

corrected qubit beyond its constituent parts98, random-ized benchmarking of logical operations404, a CNOT gatebetween two logical qubits405 as well as Ramsey interfer-ence of an encoded quantum error corrected qubit406.

Cryogenics and software development: We brieflymentioned the electrical engineering, software develop-ment, and cryogenic considerations associated with thecontrol wiring and on-chip layout of medium-scale quan-tum processors. While dilution refrigerators are nowreadily available, off-the-shelf commercial products, thedetails of how to optimally do signal-routing and rapiddata processing in a scalable fashion, is also a field inrapid development. However, with the recent demon-strations of enabling technologies such as 3D integra-tion, packages for multi-layered devices and supercon-ducting interconnects407–413, some of the immediate con-cerns for how to scale the number of qubits in the su-perconducting modality, have been addressed. On thecontrol software side, there currently exist multiple com-mercial and free software packages for interfacing withquantum hardware, such as QCoDeS414, the related py-CQED415, qKIT416 and Labber417. However, many lab-oratories use software platforms developed in-house, of-ten due to the concurrent development of custom-built,highly specialized electronics and FPGA circuits (manyof these developments are not always published, but read-ers may consult Refs. 189, 190, and 418 for three exam-ples). There is currently also a large ongoing develop-ment of quantum circuit simulation and compiling soft-ware packages. Packages such as Qiskit419, Forest (withpyQUIL420), ProjectQ421, Cirq422, OpenFermion423, theMicrosoft Quantum Development kit424 provide higher-level programming languages to compile and/or optimizequantum algorithms. For a recent review and compari-son of these different software suites, we refer to Ref.425and Ref.182 for a general review on advances in design-ing quantum software. Since the connectivity and gateset of quantum processors can differ, details of the gatecompilation implementation is an important non-trivialproblem for larger-scale processors. We note that someof these software packages already interface directly withquantum processors that are available online, supplied,for example, via Rigetti Computing or the IBM Quan-tum Experience.

Quantum error correction: While the qubit lifetimesand gate fidelity have improved dramatically in the lastdecades, there remains a need for error correction toreach large-scale processors. While certain strategies ex-ist to extend the computational reach of current state-of-the-art physical qubits426, for truly large-scale algorithmsaddressing practical problems, the quantum data willhave to be embedded in an error-correcting scheme. Asbriefly mentioned in Sections IV F 2 and IV G 2, certaincomponents of the surface code quantum error correctingscheme have already been demonstrated in superconduct-ing qubits (see e.g. Refs.66, 257, and 284). However,the demonstration of a logical qubit with greater life-time than the underlying physical qubits, remains an out-

standing challenge. While the surface code is a promis-ing quantum error correcting code due to its relativelylenient fault tolerance threshold, it cannot implement auniversal gate set in a fault-tolerant manner. This meansthat the error-corrected gates in the surface code need tobe supplemented, for example, with a T gate, to becomeuniversal. Such gates can be implemented by a tech-nique known as magic state distillation427. The processof gate-teleportation, a pre-cursor to magic state distilla-tion, has already been demonstrated using FPGA-basedclassical feedback with planar superconducting qubits190,but showing distillation and injection into a surface codelogical state remains an open challenge. The develop-ment of new quantum codes is also a field in rapid devel-opment, and the reader may consult a recent review formore details e.g. Ref.183. Another important step to-wards large-scale quantum processor architecture is thatof remote entanglement, enabling quantum informationto be distributed across different nodes of a quantum pro-cessing network428,429.

Quantum computational supremacy: Finally, we men-tion one of the grand challenges for superconductingqubits in the coming years: the demonstration of quan-tum computational supremacy430. The basic idea is todemonstrate a calculation, using qubits and algorithmicgates, which is outside the scope of classical comput-ers (assuming some plausible computational complexityconjectures). For a recent review article, the reader isreferred to Ref.431. A first step towards an approachto demonstrating quantum supremacy was recently re-ported, using 9 tunable transmons269. It is expectedthat with somewhere between 50-100 qubits432, an ex-tension of the protocol from Refs.269 and 433, will al-low researchers to sample from a classically intractabledistribution, and thereby demonstrate quantum compu-tational supremacy. The success of this program wouldconstitute a phenomenal result for all of quantum com-puting.

ACKNOWLEDGMENTS

The authors gratefully acknowledge Mollie Kimchi-Schwartz, Jochen Braumuller, and Niels-Jakob Søe Loftfor careful reading of the manuscript and Youngkyu Sungfor use of his time-dependent qubit drive simulation suiteand useful feedback from the entire Engineering Quan-tum Systems group at MIT. The authors also acknowl-edges fruitful discussion with Anton Frisk Kockum, AnitaFadavi Roudsari, Daryoush Shiri, and Christian Krizan.

This research was funded in part by the U.S. Army Re-search Office Grant No. W911NF-14-1-0682; and by theNational Science Foundation Grant No. PHY-1720311.P.K. acknowledges partial support by the WallenbergCentre for Quantum Technology (WACQT) funded byKnut and Alice Wallenberg Foundation. M.K. gratefullyacknowledges support from the Carlsberg Foundation.The views and conclusions contained herein are those of

55

the authors and should not be interpreted as necessarilyrepresenting the official policies or endorsements of theUS Government.

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