For Course of HUST Only

Simulating the Physics:Quantum Simulations in a Nutshell

Jia-Qi Cai∗

School of physics, Huazhong University of Science and Technology(Dated: March 10, 2018)

Thanks to the developing technique of full control of a quantum system, quantum simulationsand quantum computation, firstly put forward by Richard Feynman, are nowadays experimentallyfeasible to discover new physics in condensed matter physics, AMO physics and chemistry. Quantumsimulator can be implemented as a simpler quantum computer or more precisely, analogy one. Thisreview is to briefly introduce the recent progresses in quantum simulations of topological insulators,many body phenomenon, quantum optimal control and some works of mine.

PACS numbers: Quantum Computation, Quantum Control, Circuit Quantum Electrodynamics

Contents

I. Introduction 1

II. General Ideas on Quantum Simulations 2

III. Analog Quantum Simulation: a case study 3

IV. Digital quantum simulation:A case study 5

V. Conclusion 6

References 6

I. INTRODUCTION

Quantum simulators serve as controllable, easily acces-sible quantum systems to simulate much less controllablelarge system. Back to thirty years ago, to test quantumdescription of matter become a challenge to computationphysicists. It’s obvious that a large quantum system willconsume unaffordable storage. The state is described byexponentially growing numbers of parameters while thescales of the system grows linearly. One possible way toovercome this difficulty is to use approximation methodsuch as quantum Monte Carlo simulations. However,there exists so-called ”signed” problem that restricts thisefficient method. Another possible way is to use tensornetwork representation of many body wave function. Butit is shown ineffiient when treating gapless excitations.

From Richard Feynman’s great insight of theproblem[1, 2], he proposed a new type of computer: Letthe computer itself be built of quantum mechanical ele-ments which obey quantum mechanical laws.But he wasnot very specific about how can his quantum mechanicalcomputer function and be realized. About one decadelater, an ensemble of qubits serving as universal quan-tum simulator are proposed by Lloyd. In his scheme,

∗Electronic address: caidish@hust.edu.cn

the qubits can be initialized, measured and controlled bysome engineering procedure.

The potential platforms for quantum simulations andcomputations includes defects in solid system(NV cen-ter), trapped ions, neutral atoms, cavities, photonics,photons, quantum dots, superconducting circuits andNuclear spins. Based on their different advantages anddisadvantages, they can accomplish different types oftasks. For example, because of the scalability of ultra-cold atoms, the platform can be regarded as a good oneto simulate many body phenomenon. Due to the progressof controlling quantum states and scaling superconduct-ing circuits, this platform can be reagared as a good oneto achieve quantum computation. Other platforms canalso find their positions in quantum metrology, quantumsensing, precision measurement, chemistry, biology andmedicine.

To prepare ourselves with technique which will notonly benefit the physics community but also other per-spectives in normal life, we should not be fulfilled by theseachievements mentioned above. By comparing the quan-tum simulation results and some exactly solvable models,we are confident about the quantum nature of the sim-ulator and then we should proceed to build a generalquantum computer. By physicists’ efforts and innova-tions, the quantum computers are supposed to be fasterthan the summary of the computers all over the worldin at least one significant practical problem to show so-called Quantum Supremacy.

In general, there is still a long way to go for physiciststo build and test a universal computer and apply it insome practical problems. However, quantum simulationsof some physics model make great progress. For exam-ple, it’s very hard to analytically calculate the proper-ties of many body Hamiltonians with interaction, such asMott-Hubbard Hamiltonian, t-J Hamiltonian and quan-tum spin chain Hamiltonian. Many interesting quantumphases , entanglement features and topological propertiesare hidden deeply in the nature of many-body Hamilto-nians. To fully extract informations from those systemsrequires a simulator of quantum nature.

The paper is organized as below: I will briefly intro-duce some rather general quantum simulation problems

2

FIG. 1: Scheme of quantum simulations. The original sys-tem in the gray block is less accessible and controllable. Awell-defined simulator means that its initial state can be pre-pared, the evolution of the state can be engineered, and thefinal state can be measured. After mapping from the originalsystem to the simulator and performing quantum gates , onecan extract information from simulator. For the two systemobey the same physical law, the information one obtained canbe remapped to original system.

in SectionII. Then I will focus on quantum simulationsof some topological band theory by ultracold atoms andsuperconducting circuits in SectionIII. A recent work onqubits control by holonomic quantum gates with trans-mon qubits is introduced as an example of proceedingfrom quantum simulator to quantum computer in Sec-tionIV .

II. GENERAL IDEAS ON QUANTUMSIMULATIONS

Definition of Quantum Simulations We first considera less controllable system with Hamiltonian Hsys and de-note the states in the target Hilbert space as |ψ〉sys. Asquantum mechanics tells, evolution operator from the ini-tial state |ψ (t0)〉sys to the final state |ψ (t)〉sys is unitary

and denoted by U (t, t0) = T exp [−i~Hsys (t− t0)],where T is the time-ordering operator. And a more con-trollable system evolves from |φ (t0)〉sim to |φ (t)〉sim via

U (t, t0) = T exp [−i~Hsim (t− t0)]. Then if a map-ping from the original system to the quantum simulatorexists, the system can be simulated as shown in FIG 1.

In a more algebraic language, the representation of asimulator Hamiltonian can be constructed as :

Hsim/sys = H0 +Hd (t)

H0 =∑

i,j,m,n

vi,jm,n

(a†i

)majn

Hd =∑

i,j,m,n

ui,jm,n (t)(a†i

)majn

(1)

Where ai, a†i denote bosonic or fermionic annihilation

and creation operators, respectively. H0 is the simula-tor Hamiltonian which is fixed and Hd is the artificial

FIG. 2: A typical superconducting circuit’s excitation spec-trum . The system is capable for constructing a single qubitfor the large energy gap between the subspace of 1st,2nd ex-cited states and others

Hamiltonian which can be modulated so the system ismore controllable. Most models of the simulators andoriginal systems can be easily represented by this lan-guage. To simulate a quantum system, the dimensionof the Hilbert space of simulator must be larger thanthe original system. Then, preserving the commuta-tion relations of ai and aj , one can take the injection

ϑ : v (u)i,jm,n → v (u)

i,jm,n.

Classifications of Quantum Simulations There actuallyexists several types of quantum simulation because thedifferent idea used in modulating the Hamiltonian andsimulating procedure. Digital quantum simulation(DQS)only require single-qubit gate and two-nontrivial-qubitgate to achieve quantum simulation. The wave func-tion are encoded by the computational qubits and anycomplicated many-qubit unitary transformation can beimplemented through the application of a sequence ofsingle- and two-notrivial gates. Since DQS has the capa-bility to simulate any unitary transformation, the DQSis universal but not all unitary transformations can besimulated with polynomial resources which led to fidelityproblems. Therefore DQS can be only treated as a sim-pler quantum computation method. The components ofthese simulator often involve in selecting an effective two-level system to construct a qubit as shown in FIG 2. Be-cause of the large gap between qubit and other energylevel, the Markovian approximation is valid and otherenergy and influence from environment can be treatedas damping and dephasing together. In a more specificexample,e.g, transmon qubit, I will show how to reduce

3

FIG. 3: An algorithms(Quantum Support Vector Machine)implemented by a sequence of quantum gates.(Digital quan-tum simulation or quantum computation)

crosstalk among subspace and other enegy levels usingquantum optimal control. Another question is the influ-ence of local perturbation or error induced by uncertaintyfrom controller, e.g laser or microwave generator., whichmeans: Hd (Ω (t)) → Hd (Ω (t)± ε). Some of them canbe avoided by proper choice of quantum gate implemen-tation, as will be shown in section IV[1].Analog quantumsimulation(AQS) is a natural but not universal methodin which the simulator mimics another system. An un-controllable Hamiltonian can be directly mapped to an-other one which can be controlled to some extent. Unlikeestablishing units in DQS, we treat AQS as a natural sys-tem with effective many body Hamiltonian. So even ifthe system is less controllable in some special parame-ter, the phase transitions under study can be sometimesobserved and thus provide the question to several typesof many-body phenomenon. In the standard formalismof AQS, one must first find the mapping between targetsand the simulator. Finding a mapping seems simplerthan obtaining a sequence of gates for a given Hamilto-nian. But actually sometimes the finding the correct andclever mapping is complicated and need devising.

III. ANALOG QUANTUM SIMULATION: ACASE STUDY

I will present a practical example to illustrate the basicidea of analog quantum simulation. I propose an experi-mentally feasible scheme to observe the Berry curvatureand Chern number in a topological transition process, us-ing an artificial three-level atom. The Berry curvature isdefined in a two dimensional manifold with an adjustablephase factor. By varying parameters of the system slowlybut non-adiabatically, Berry curvature can be measuredby dynamic hall effect. The simulation shows a sign re-versal of Chern number from continuously changing thephase factor, which indicates special topological transi-tion.

Berry curvature[3], which is analogous to the magneticfield but in a parameter space may contribute a lot tostudy topological properties and provide a new perspec-tive of gauge field. Based on the concept of Berry cur-vature,it is discovered that topological invariants such asChern number exists in many typical quantum systemssuch as condensed matter, photonics and quantum cir-cuits. [3–6]

1

0

e

1ω

eω ( )1 2

cosieφ α α+

( )1cosi

eφ α−

( )2cosieφ α

FIG. 4: Scheme of ∆-type atom.

Meanwhile, in recent years, Berry curvature is mea-sured by transport methods with solid-states, ultracoldatoms and photonics.[7, 8] Compared with anomaloustransport methods, a circuit quantum dynamics simula-tor can improve integration and scalability. The advancesof a long dephasing time and good controllability lead toa success in measuring Berry curvature and Chern num-ber with an artificial two-level atom made by quantumcircuits.[9–12] But lots of related physics remain to bediscovered in a reduced 3D Hilbert space, for example,Kagome lattice problem. Motivated by this, we proposea single artificial ”∆”-type three-level atom to simulatethe Kagome lattice, which shows interesting physical phe-nomenon such as flat band and topological transitions.

In our scheme, we consider 2D manifold of parame-ters in addition to the 0D system of a single atom, inorder to simulate a long-ordered and ideal Kagome lat-tice Hamiltonian and obtain Berry curvature and furtherChern number. We go beyond level counting method inthe Hofstadter spectrum. We use dynamical hall effectof a single atom with varying parameter which has alsoan intuitive interpretation of simulated Lorentz force inthe parameter space in presence of Berry curvature.

We consider a cycle-three-level atom in FIG.?? witharbitary transitions permitted. The atom has two groundstates, |0〉 and |1〉 and an excited state |e〉. We beginwith this Hamiltonian to demonstrate the essential stepsto measure Berry curvature:

H = ωe |e〉 〈e|+ ω1 |1〉 〈1|+ eiφ cos (α1 + α2) |e〉 〈0|+ e−iφ cos (α1) |e〉 〈1|+ eiφ cos (α2) |0〉 〈1|+ h.c.

(2)

This system is dependent on three different param-eters: H = H (α1, α2;φ). Under the transformationα1,2 → α1,2+2π, the Hamiltonian remains invariant. Weconstraint these parameters in the zone: −π 6 α1,2 < πand hence the full manifold is torus-like. As discussedin the Appendix, Berry curvature and Chern number aredefined in this closed manifold.

Then, the Hamiltonian can be simply diagonalized and

4

a b

c d

FIG. 5: The energy band of the system in the stimulatedmanifold when flat band occurs. The energy bands are pro-jected to α1 − E plane. (a)φ = 0 and ω1 = ωe = 0, thesystem is gapless and a band is made flat. (b)φ = 0 andω1 = 0, ωe 6= 0, the first band becomes dispersive. (c)φ = π

6and ω1 = ωe = 0, the system is gapped and the mid band ismade flat. (d) φ = π

6and ω1 = 0, ωe 6= 0 but ω 1, the mid

band becomes dispersive.

the eigen-energies can be obtained:

E1E2E3 = E1cos2α2 + E2cos2α1 + E3cos2 (α1 + α2)

+ 2 cosα1 cosα2 cos (α1 + α2) cos 3φ(3)

Where E1 (α1, α2) = E (α1, α2) − ωe,E2 (α1, α2) =E (α1, α2) and E3 (α1, α2) = E (α1, α2) − ω1 andE(α1, α2) is the energy. Note that for fixed α1,2 and φ,there are three values of E(α1,2). With the definition ofthe closed manifold, this can be interpreted as an energyband.

In our scheme, there are some biases in the secondand third diagonal matrix element of the Hamiltonianω1 and ωe, respectively. But in a special frame, whenω0 = ω1 = ωe = 0, from Eq.3, we obtain:

E3 = (2t+ 1)E + t cos (3φ)

t = 2 cosα1 cosα2 cos (α1 + α2)(4)

The middle band will be flat when the conditionφ = π

6 + nπ3 is satisfied. The upper or lower band will be

flat when the condition φ = nπ3 is satisfied. Some special

cases are shown in FIG 5. When ω1,e = 0, the lower bandand mid band can be made fully flat. A small perturba-tion of ω1,e lead to dispersive bands but these bands arestill relatively flat. By tuning φ, with ω1,e 1, finitegaps are opened. When ω1,e is large enough, the gapsare closed and degeneracy occurs. In our method, weonly consider the former case because when degeneracyoccurs or the gap is so small, there is large deviationsand the method lose its efficiency. [13]

Mapping to the Kagome Lattice. In the bosonic rep-resentation of the Hamiltonian Eq.2, it can be written

FIG. 6: Reconstruction of Berry curvature by dynamic simu-lation and theoretical numerical calculation. In the dynamicsimulation, we choose four different configuration of veloc-ity and find the slop of the response-velocity curve to obtainBerry curvature: ω2

α = 0.01, 0.04, 0.09, 0.16

as:

H (α1, α2)

= ψ†

0 eiφ cos (α1 + α2) e−iφ cos (α1)ω1 eiφ cos (α2)

ωe

ψ(5)

Where ψ = (ψa, ψb, ψc) is the vector of bosonic destroyoperator. By inspecting the Hamiltonian above and theHamiltonian of Kagome lattice, we find that they mapto each other by a transformation: α1 → −k2, α2 →k3, α1 + α2 → −k1, H (α1, α2) = −2gH (−k2, k3) . Thehopping matrix element and accumulated phase of hop-ping from one site to the nearest one is denoted by gand φ, respectively. ω1,e denote the anisotropy of thesublattice.[14, 15]

Topological Invariant.We now consider how to measureBerry curvature and Chern number directly and apply itto our model to discuss how they change with differentvalues of parameter.

Experimental and theoretical method. A time-dependent parameter and a dynamical process are neededto obtain Berry curvature. For instance, we simply setα1 = α1(t) = ω2

αt2 to demonstrate the method. The ini-

tial state is the nth eigenstate of the Hamiltonian whereα2 and φ is fixed and α1(0) = 0. Then, the system evolvesa certain time tf and then a linear response can be mea-sured . The Berry curvature of the point (α1(tf ), α2) canbe extracted by :

〈Fµ〉f = 〈Fµ〉0 + Ωnµλdsλdt

+O

((dsλdt

)2)

(6)

Where the generalized force Fµ = −∂µH is along the λdirection and the velocity of the change in the param-eter is along the µ direction. We denote (α1, α2) by ~s.The expectation 〈. . .〉0 is taken in the initial nth eigen-state of the Hamiltonian and 〈. . .〉f = 〈ψ (tf )| . . . |ψ (tf )〉where |ψ (tf )〉 is the final state after an evolution due toparameter varying.[13]

5

/4 /3 /2-1.5

-1

-0.5

0

0.5

1

1.5

Ch

11,

2

Dynamic Simulation

Theoretical Caculation

FIG. 7: Sign reversal of Chern number. Dynamic simulationsgive a fine quantized Chern number with ωα = 0.01.

For the same original parameters, 〈Fµ〉0 is fixed. One

can plot 〈Fµ〉f −dsλdt curve and the slop is Berry curva-

ture. We note that by using the dynamic hall effect tomeasure Berry curvature, the quench velocity is turnedon smoothly and the system is prepared in a initial statewith large gap.

To justify the measurement of Berry curvature, onecan numerically calculate it by the definition like Eq ??but derivatives of state vectors are used. For the Hamil-tonian is not analytically diagonalizable, numerical sim-ulation creates discrete state vectors. By insert the iden-tity

∑n|ψn〉 〈ψn| = I, the Berry curvature associated with

only state vectors and derivatives of Hamiltonian can bederived:

Ωnµλ = i∑m6=n

〈ψn| ∂µH |ψm〉 〈ψm| ∂λH |ψn〉 − µ↔ λ

(Em − En)2

(7)Applying this equation to our model, we found that

Ωα1,α2−π = Ωα1−π,α2= Ωα2,α1

= Ωα1,α2. Therefore, we

can restrict the region to 0 ≤ α1,2 < π.A topological invariant named Chern number can be

extrated from the integration of Berry curvature in thewhole parameter space.

Topological Transitions. The major findings in ourmodel Hamiltonian include two types of topological tran-sitions. In our system, topological transition happensbetween two different topological phases of the system,which is characterized by Chern number.

IV. DIGITAL QUANTUM SIMULATION:ACASE STUDY

We proceed to illustrate the proceedings from quantumsimulator to quantum computer by holonomic quantumgates with transmon qubits. The origin paper of this ideahas been submitted as letter by me and my co-authors.For pedagogical purpose, I will briefly introduce what aI’ve done and thought about achieving the most impor-tant segment in digital quantum simulation-Achievingquantum gates with high fidelity. In the current im-plementation, each transmon serves as a qubit, and the

qubit states are defined as in the experiment of Ref.[16]. An arbitrary holonomic single-qubit gate can be ob-tained by varying the amplitudes and phase difference oftwo microwave fields resonantly coupled to a transmon.Meanwhile, the gate is achieved with a single-loop sce-nario [17–19], which is different from Ref. [16]. Moreover,nontrivial two-qubit gates can be induced with a trans-mission line resonator(TLR) simultaneously coupled totwo qubits driving by microwave fields, which leads toeffective resonant TLR-qubit couplings for the two in-volved qubits. In addition, our proposal can be readilyscaled up to a two-dimensional lattice for scalable quan-tum computation. Therefore, we present a simple andresonant scheme for robust digital quantum simulationon superconducting circuits with transmon qubits.

Universal single-qubit gates We now proceed to presentour scheme. We first consider the implementation ofan arbitrary single-qubit gate which consists of two mi-crowave fields with amplitudes Ωi (i = 0, 1) and initialphases φi resonant coupled to the sequential transitionsof the three lowest three levels |0〉, |e〉 and |1〉 of a trans-mon, with |0〉 and |1〉 are the qubit states. The Hamil-tonian of the system can be written as

H1 = Ω0eeiφ0 |0〉〈e|+ eiφ1Ω1e|1〉〈e|+ H.c.

= Ωei(φ1−π)(

sinθ

2eiφ|0〉〈e| − cos

θ

2|1〉〈e|

)+ H.c.

≡ Ωei(φ1−π)|b〉〈e|+ H.c., (8)

where φ = φ0 − φ1 + π, Ω =√

Ω20e + Ω2

1e, tan(θ/2) =Ω0e/Ω1e, and the bright state is |b〉 = sin(θ/2)eiφ|0〉 −cos(θ/2)|1〉. That is, the dynamics of the system is cap-tured by the resonant coupling between the states |b〉and |e〉. Therefore, when the cyclic evolution condi-

tion is met, i.e.,∫ T0

Ωdt = π, one can obtain certainsingle-qubit gate by choosing different θ and/or φ. Mean-while, an arbitrary single-qubit state can be written as|ψ〉 = α|0〉 + β|1〉 ≡ [α sin(θ/2)e−iφ − β cos(θ/2)]|b〉 +[α cos(θ/2) +β sin(θ/2)e−iφ]|d〉 with the dark state to be|d〉 = cos(θ/2)|0〉 + sin(θ/2)e−iφ|1〉, so that there is notransitions between the |d〉 and |b〉 states during the evo-lution. Therefore, the obtained single-qubit gates are ofthe holonomic nature .

In the qubit subspace, when φ1 = π, the implementedsingle-qubit gate reads

U1(θ, φ) =

(cos θ sin θeiφ

sin θe−iφ − cos θ

)= ie−i

π2 n·σ, (9)

which is a π/2 rotation of a qubit state around axisn = (sin θ cosφ, sin θ sinφ, cos θ). That is, only the fixedπ/2 rotation can be obtained. In this case, in order torealize universal single-qubit gates, at least two sequen-tially gates with different θ and/or φ are needed.

When the total evolution time T is divided into twoequal parts, i.e., φ1 = π for t ∈ [0, T/2] and φ′1 = γfor t ∈ [T/2, T ]. In this case, the Hamiltonian ofthe two parts are Ha = λ1(|b〉〈e| + |e〉〈b|) and Hb =

6

(d)(c)

(a) (b)

-0.2 0 0.20.9994

0.9995

0.9996

0.99970.9997

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Et/2

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Nt/2

0 0.2 0.4 0.6 0.8 1 1.20.2

0.4

0.6

0.8

1

N/Et/2

|0

|e

|1

F

|0

|e

|1

F

FE

FN

0.95 1 1.050.990.995

1

FIG. 8: States population and fidelity dynamics of (a) NOTand (b) entangle gates as a function of ΩN/Et/π with theinitial state being |0〉. (c) The dynamics of the NOT andentangle gaies fildelities. (d) The stability of the NOT gateto certain random fluctuations εΩ.

−λ1(eiγ |b〉〈e| + e−iγ |e〉〈b|), and the corresponding evo-lution operator are Ua = |d〉〈d| − i(|b〉〈e| + |e〉〈b|) andUb = |d〉〈d|+i(eiγ |b〉〈e|+e−iγ |e〉〈b|), respectively. There-fore, the implemented single-qubit gate will be

U1 = eiγ2

(cos γ2 − i sin γ

2 cos θ −i sin γ2 sin θeiφ

−i sin γ2 sin θe−iφ cos γ2 + i sin γ

2 cos θ

)= ei

γ2 exp

(−iγ

2n · σ

), (10)

up to a global phase factor, in the qubit subspace, whichcorresponds to a rotation around n by an angle γ and caninduce universal single-qubit gates, in a holonomic way. Geometrically, the two Hamiltonian corresponds twodifferent loops in the Bloch sphere, but the final point (atT/2) of Ha is in coincidence with the start point (at T/2)of Hb, the final point (at T ) of which is in coincidencewith the start point of Ha (at 0). That is, the two loopscoincidence at 0 and T/2, and we use the slice contourfor the cyclic geometric phase.

We now proceed to the implementation of nontriv-ial two-qubit gates, where we consider the case thatthe two driving qubits are coupled simultaneously tothe TLR.both the TLR and the driving microwave fieldare dispersively coupled the transitions |0〉 ↔ |e〉 and|1〉 ↔ |e〉, with the coupling strength for ith qubit be-ing gi and Ωi and their corresponding frequencies to be

ωc and ωi, respectively. Meanwhile, the two couplingfroming a two-photon resonant situation, i.e., ωc−ω0e =ω1e−ωi = ∆ > α , ω0e = Ee−E0, and ω1e = E1−Ee, andthus leads to driving assisted coherent coupling betweenthe TLR and the |0〉 ↔ |1〉 transition . When ∆ gi,

t T t T 2T

time0.0

0.2

0.4

0.6

0.8

1.0

stat

e po

pula

tions (40,0.990)

|100|001|010Loss

FIG. 9: States population and fidelity dynamics for gates asa function of time with the initial state being |100〉.

the interacting system can be written as

H2 =

2∑i=1

gi(e−iϕia|1〉i〈0|+ H.c.), (11)

where gi =√

2giΩiα/[∆(∆−α)] and ϕi being the initailphase of the microwave driving field on ith qubit.

As shown in FIG.9, the rising time of the pulse ∆t is14T and the effective interaction strength is g = 10Mhz.Our implementation can achieve more than 99% fidelityrate and become a good platform for digital quantumsimulation.

V. CONCLUSION

In this review of quantum simulation, I discuss on twotype of quantum simulation problem briefly. The mainproblem of analog quantum simulation is how to map theoriginal system to our quantum simulator, as a kind ofsolution illustrated by topological insulator. The mainproblem of digital quantum simulation is how to con-trol the quantum simulator and achieve higher fidelity ofquantum gates, as a kind of solution illustrated by holo-nomic quantum gates with transmon qubits.

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