protecting quantum superpositions in josephson …

PROTECTING QUANTUM SUPERPOSITIONS

IN JOSEPHSON CIRCUITS

Acknowledgements to Yale quantum information team members:

M. Hays

S. Mundhada

A. Narla

K. Serniak

V. Sivak

K. Sliwac

S. Touzard

R. Heeresd

Y. Liu

W. Pfaff

N. Ofek

U. Vool

C. Wangf

Z. Wang

G. de Lange

A. Grimm

A. Kou

Z. Leghtasb

M. Hatridgea

S. Shankar

L. Frunzio

S. Girvin

L. Glazman

L. Jiang

M. Mirrahimi

R. Schoelkopf



C. Axline

J. Blumoff

K. Chou

Y. Gao

A. Petrenkoc

P. Rheinhold

A. Royf

Now at: aU. Pittsburgh, bENS Paris, cQCI, dCEA Saclay, fU. M. Amherst, fU. Aachen

Sponsors:

Michel Devoret, Yale University

Capri Spring School 2017

LECTURE I: INTRODUCTION AND OVERVIEW



LECTURE II: CAT CODES

LECTURE III: PARITY MONITORING AND CORRECTION

PROTECTING CLASSICAL INFORMATION

1

0

0 1

ONE BIT OF INFORMATION

ENCODED IN A SWITCH

ENERGY BARRIER

PROTECTS INFORMATION

energy of switch

switchposition

BU k T

error expB

U

k T

ELEMENT OF QUANTUM INFORMATION: QUBIT

X

Y

Z

MUST PRESERVE ALL SUPERPOSITIONS, I.E. THE ENTIRE SPHERE

Bloch sphere

cos 0 exp sin 12 2

i

0 1

Two basis quantum states:

Generic superposition:

CARDINAL POINTS OF THE BLOCH SPHERE

X

Y

Z

1

1

1

1

1

1

Z

Z

X

X

Y

Y

DECOHERENCE:

LOSS OF

INFORMATION

X

Y

"T1" "Tf"

Z

BY CAREFUL ENCODING IN A LARGE ENOUGH SPACE,

QUANTUM INFORMATION CAN BE PROTECTED

ELEMENTARY ERRORS E ONLY MOVE OCTAHEDRON ALONG A 3+nth DIMENSION, WITHOUT CHANGING ITS VOLUME

X Y

Z

E

SIMPLE EXAMPLE: IN GENERAL:

cos e sin2 2

i

X AND Y COMPONENTS IN M = 0

MANIFOLD ARE PROTECTEDFROM NOISE IN COMMON MODE

MAGNETIC FIELD B

3 STRATEGIES FOR PROTECTING QUANTUM INFORMATION

- Encode into a quantum cavity resonator and monitor photon number parity

- Encode into a register of qubits, monitor multi-qubit parities

Examples: Steane & surface codes

Example: cat codes

- Encode using defects in topological quantum matter

Example: Majorana fermions

1

1

1

1

1

1

1

0

1

1

1

0

0

0

1

1

0

0

0

0

1

0

1

0

0

1

1

0

1

0

0

1

1

0

0

0

0

1

0

1

0

1

0

0

0

1

0

1

1

1

0

1

0

0

1

0

+ + + +

+++

0

0

0

0

0

0

1

0

0

0

1

1

1

0

0

1

1

0

1

0

1

1

0

0

1

0

1

1

0

0

1

1

1

1

0

1

0

1

0

1

1

1

0

1

0

0

0

1

0

1

1

0

1

+ + + +

+++

0

1

1

0L

1L

REGISTER ERROR CORRECTION: STEANE CODE

0 1 2e e eX

0 1 2e e eX

0 1 2e e eX

0 1 2e e eX

0 1 2e e eX

0 1 2e e eX

0 1 2e e eX

3 4 5e e eZ

3 4 5e e eZ

3 4 5e e eZ

3 4 5e e eZ

3 4 5e e eZ

3 4 5e e eZ

3 4 5e e eZ

STEANE CODE ERROR CORRECTION PROTOCOL

i = 0Z Z Z

Z Z

Z Z

ZZ

Z

Z

Z

X X X

X X

X X

XX

X

X

Xi = 6

ej

j = 0 j = 5

qi

in cosA t A t

out cosA t A t f

SUPERCONDUCTING QUBIT PARADIGM:

CONSIDER A SINGLE, DRIVEN ELECTROMAGNETIC MODE...

0

0 DRIVE SIGNAL

REFLECTED SIGNAL

MICROWAVE FREQUENCIES0 Bk T

Hydrogen atomSuperconducting

LC oscillator

ATOM vs CIRCUIT

L C

metallic latticepositive ions

electron condensate


LC oscillator

L C

ATOM vs CIRCUIT

metallic latticepositive ions

electron condensate


LC oscillator

L C

unique electron → whole superconducting condensate velocity of electron → current through inductorforce on electron → voltage across capacitor

ATOM vs CIRCUIT


LC oscillator

L C

unique electron → whole superconducting condensate velocity of electron → current through inductorforce on electron → voltage across capacitor

ATOM vs CIRCUIT

1mm0.1nm

+Q

F

-Q

F

E

LC CIRCUIT AS A QUANTUM

HARMONIC OSCILLATOR

r

L C

ˆˆ ,Q i F

F = LI

Q = CV

F

F

WAVEFUNCTIONS OF LC CIRCUIT

rI

E

YF

Y0

0Y1

In every energy eigenstate,(microwave photon state)current flows in opposite directions simultaneously! F

F

F

EFFECT OF DAMPING

important: as little

resistance as possible dissipation broadens energy levels

E

11

2 2n r

r

iE n

RC

F

F

CAN PLACE CIRCUIT IN ITS GROUND STATE

r Bk T10-5 GHz 10mK

rI

E

some cold dissipation isactually needed: provides

reset of circuit!

UNLIKE ATOMS IN VACUUM,

TWO CIRCUITS EASILY COUPLE

THROUGH THEIR WIRES:

COUPLING CAN BE ARBITRARILY STRONG!

HAMILTONIAN BY DESIGN!

F

F

E

CAVEAT: THE QUANTUM STATES OF A

PURELY LINEAR CIRCUIT

CANNOT BE FULLY CONTROLLED!

NO STEERING TO AN ARBITRARY STATE

IF SYSTEM PERFECTLY LINEAR

r

Potential energy

Position coordinate

NEED NON-LINEARITY TO FULLY

REVEAL QUANTUM MECHANICS

Potential energy

Position coordinate



Emissionspectrum@ high T

frequency

01122334

0

1

2

3

4

Potential energy

Position coordinate



Emissionspectrum

frequency

01122334

absolutenon-linearity

is ratio of peakdistance topeak width

0

1

2

3

4

JOSEPHSON TUNNEL JUNCTION:

A NON-LINEAR INDUCTOR

WITH NO DISSIPATION

1nmS

I

S

superconductor-insulator-

superconductortunnel junction

CJLJ

100 -1000 nm

MIGHT BE SUPERSEDED SOME DAY BY ANOTHER DEVICE(BASED ON ATOMIC POINT CONTACTS, NANOWIRES, ???)



WITH NO DISSIPATION

1nmS

I

S



f

II f / LJ

0 0sin /I I f f

CJLJ

I

' 't

V t dtf

02e

f

0

0

JLI

f

0I

1nmS

I

S



f

0cos /JU E f f

CJLJ

I

' 't

V t dtf

02e

f

2

0J

J

LE

f

bare Josephson potential

2 JE



WITH NO DISSIPATION

Potential energy

Position coordinate

ELECTROMAGNETIC OSCILLATOR ENDOWED WITH JOSEPHSON

NON-LINEARITY

Emissionspectrum@ high T

frequency

01122334

0

1

2

3

4

Kerr frequency

QUANTUM STATES OF A HARMONIC OSCILLATOR

REPRESENTED BY THEIR QUASI-PROBABILITY WIGNER FUNCTION IN PHASE SPACE

I

Q

vacuum: |n=0>

I

Q

squeezed vacuum

I

Q

coherent state

I

Q

cat state

I

Q

1 photon: |n=1>

2

2

†

; 1a a

W D PD P

I iQ

I

Q

vacuum: |n=0>

I

Q

squeezed vacuum

I

Q

coherent state

I

Q

cat state

I

Q

1 photon: |n=1>

2

2

QUANTUM STATES OF A HARMONIC OSCILLATOR

REPRESENTED BY THEIR QUASI-PROBABILITY WIGNER FUNCTION IN PHASE SPACE †

; 1a a

W D PD P

I iQ

0L

1L

i i

' '

' 'i i i

CAT ENCODING, AND INFORMATION DECAY

even

even

odd

odd

typical

time

evolution

3

a

1t

BLOCH SPHERE OF

4-LEG CAT CODE

Leghtas et al. PRL (2013)Vlastakis et al. Science (2013)Mirrahimi et al. NJP (2014)Sun et al. Nature (2014)

+Z

+X

+Y

TWO TYPES OF CAT ERROR MUST BE ADDRESSED:

- COHERENT STATE SHRINKING, DEPHASING

& DEFORMING

- PHOTON NUMBER PARITY JUMPING

HOW DO WE FULLY CONTROL THE RESONATOR?

DEVICES

2 mm

0.5 mm

1.2

cm

90 μm3.4 cm

N. Bergeal et al. Nature (2008)

H. Paik et. al. PRL (2011)

G. Kirchmair et. al. Nature (2013)

JosephsonRing Modulator

Al

Si

EXPERIMENTAL SETUP

instrument rackdilution refrigerator

microwave

generators

atomic clock

arbitrary

waveform

generator

acquisition

card

77 K

100 mK

1 K

4 K

quantum

circuits

20 mK

0.5 m

amplifier

electronics

sL

2 OSCILLATORS COUPLED

THROUGH A TUNNEL

JOSEPHSON JUNCTION

TE101 modes

E

storagedump

&

readout

junction

,

0

, ,r s r s r sL C 2 2 2 2

0cos /2 2 2 2

s s r rJ r s

s s r s

Q QH E

C L C Lf

F F F F

sC rLrC

SIMPLIFIED MODEL

A POWERFUL DETERMINISTIC GATE

COMBINING MICROWAVE LIGHT AND A QUBIT

cavity oscillator

superconducting qubit

†

zH a a

1

2c q T

strong dispersive regime: Stark shift per photon > 1000 linewidths

Can combine two cavities, a qubit and a quantum limited Josephson amplifier:

Sstorage cavity

superconducting qubit

readout cavity

S

R

Leghtas et al., Phys. Rev. A87, 042315 (2013)

amplification chain

s


THROUGH A TUNNEL

JOSEPHSON JUNCTION

TE101 modes

E

storagedump

&

readout

junction

rq


THROUGH A TUNNEL

JOSEPHSON JUNCTION

† † †/ s s s r r r q q qH a a a a a a -cqq

2aq

†( )2

aq( )2

- cqsaq

†aqas

†as - cqraq

†aqar

†ar + ...

TE101 modes

E

storagedump

&

readout

junction

-crs

4as( )

2ar

†( )2

+ h.c.+ ...

NON-DEMOLITION MEASUREMENT

OF SUPERCONDUCTING ATOM

Readout phase

f

90º

-90º

fc

resonator linewidth ~ c

q ~ p /2

Readout amplitude

f

fc

dispersive shift c

JPC

e

e

g

g

HEMT-amp

I

Q

2 2I Q

1tan /Q I

Circuit QED: Wallraff et al. (2004),

Houck et al., (2007), Gambetta et al. (2008),

Vijay et al., (2012).

ref

readout

frequency

qubit + resonator

Josephson

pre-amp

Cavity QED: Pioneering expts in

Haroche's and Kimble's groups

0

5

-5

0 50 100 150Time (ms)

I m/σ

7.5σ

~ 90% of fluctuations ofsignal at 300K are quantum noise!

~2000 bits per qubit lifetime

QUANTUM JUMPS OF QUBIT STATE

TWO TYPES OF CAT ERRORS MUST BE ADDRESSED:


& DEFORMING


REVIEW THE USUAL DRIVEN-DAMPED

HARMONIC OSCILLATOR

x

p


HARMONIC OSCILLATOR

H = e1

*a + e1a†

k1a

Hamiltonian :

loss operator :

a = -2ie1 /k1

Re(β)

Im(β)

Wigner function W(β)


HARMONIC OSCILLATOR

AVERAGE EVOLUTION OF DENSITY MATRIX FOR AN OPEN SYSTEM

† † †

,

1

2

di H c

dt

c c c c c c c

loss operator

A SPECIAL DRIVEN-DAMPED OSCILLATOR


Force ~ X2


Force ~ X2

Damping

~ V2


Re(β)

Im(β)



H = e2

*a2 + e(a†)2

k 2 a2

Hamiltonian :

loss operator :

a = ± -2ie2 /k 2

Re(β)

Im(β)



H = e2

*a2 + e(a†)2

k 2 a2

Hamiltonian :

loss operator :

a = ± -2ie2 /k 2

Re(β)

Im(β)


Damping

Force 2X

2X

Mirrahimi et al., New J. Phys. 16, 045014 (2014)

s


THROUGH A TUNNEL

JOSEPHSON JUNCTION

TE101 modes

E

storagedump

&

readout

junction

rq


THROUGH A TUNNEL

JOSEPHSON JUNCTION

† † †/ s s s r r r q q qH a a a a a a -cqq

2aq

†( )2

aq( )2

- cqsaq

†aqas

†as - cqraq

†aqar

†ar + ...

TE101 modes

E

storagedump

&

readout

junction

-crs

4as( )

2ar

†( )2

+ h.c.+ ...

p 2 s d

drive pump

H

sr= g

2

*as

2ar

† + h.c.( )

effective loss operator:

k2a

s

2effective drive Hamiltonian:

e

2a

s

†( )2

+ h.c.

TWO-PHOTON EXCHANGE BETWEEN THE 2 CAVITIES

storage

ω

dump

ωωs

ωp

ωs

ωd

ωs

ωs

ωp

d

density

of states

CAVITY BI-STABLE BEHAVIOR (SEMI-CLASSICAL)

Re(β) = I

Im(β) = Q


π/2 W(β)

Re(β)

Im(β)


π/2 W(β)

Re(β)

Im(β)

Re(β)

Im(β

)

-2 0 2

-2

0

2

π/2 W(β)

Re(β)

Im(β

)

-2 0 2

-2

0

2


π/2 W(β)

Re(β)

Im(β

)

-2 0 2

-2

0

2


see also Puri and Blais arXiv:1605.09408

pumping time

raw data

reconstruction

numerical

simulation

0 μs 2 μs 7 μs 19 μs

W(β)

Re(β)

Im(β

)

-2 0 2

-2

0

2

quantum

interference !

CAT SQUEEZES OUT OF VACUUM !

pumping time

raw data

reconstruction

numerical

simulation

0 μs 2 μs 7 μs 19 μs

W(β)

Re(β)

Im(β

)

-2 0 2

-2

0

2

quantum

interference !

CAT SQUEEZES OUT OF VACUUM !

Data of two years ago…

Leghtas et. al. Science (2015)

M. Reagor et al., APL (2013), C. Axline et al. APL (2016)

Wigner qubitseparate fromstorage ancilla

0.0

0.3

0.6

-0.3

-0.6

W

Im(𝛽

)

0

-0.5

0.5

-0.5 0.50Re(𝛽)

NOW: 𝜅2

𝜅1= 100 with 1/𝜅1 = 92 μs

PRESENT “CAT-PUMPING” COHERENCE

Touzard, Grimm, et al. arXiv:1705.02401

Im(𝛽

)

-0.5 0.50

0 0.0

0.3

0.6

-0.3

-0.6

Re(𝛽)

-0.5

0.5

W

+ 𝛼| − 𝛼⟩

0 L =𝛼 + −𝛼

2

1 L =𝛼 − −𝛼

2

RABI OSCILLATIONS OF CAT WHILE PUMPING

observation of quantum Zeno dynamics of protected information!

Movie slowed down by 10-6

Misra & Sudarshan (1977); Itano et al., (1990); Fisher, Gutierrez-Medina & Raizen (2001); Facchi & Pascazio (2002); Raimond et al., (2012)

0.0

0.5

1.0

-0.5

-1.0

W

Re(𝛽)

Im(𝛽

)

-2 20

0

1

-1

MEASUREMENT OF THE 4 PAULI OPERATORS OF THE PROTECTED

QUBIT MANIFOLD

log

Xπ/2log

+X

+Y

+Z

+XL

+YL

+ZL

+XL

+YL

+ZL

/ : N( 𝛼 ± −𝛼 )

/ : N( 𝛼 ± 𝑖 −𝛼 )

/ : ±𝛼

PROCESS TOMOGRAPHY OF QUANTUM ZENO RABI ROTATION

Encode

Identity

+X

+Y

+Z

Zeno 𝜋/2 rotation around X

Xπ/2

+X

+Y

+Z

DURATION OF LOGICALGATES: 1.8 ms

GATE DONEWHILECORRECTING!

TWO TYPES OF CAT ERRORS MUST BE ADDRESSED:


& DEFORMING


transmon

cavity

transmon cavity

SYNDROMEMEASUREMENT

transmon

cavitySYNDROMEMEASUREMENT

transmon cavity

evenn

oddn

transmon


transmon cavity

transmon

cavity

evenn

oddn

SYNDROMEMEASUREMENT

transmon cavity

transmon


transmon cavity

evenn

oddn

transmon

cavity

transmon cavity

evenn

SYNDROMEMEASUREMENT

REAL-TIME QUANTUM CONTROLLER FOR FEEDFORWARD CORRECTION

Analog

Outputs

Analog

Inputs

FPGADACs

ADCs

Bare Transmon 4Cat Code NO QEC 4Cat Code W/ QEC

Qubit after Qubit after

Fidelity = 0.58 Fidelity = 0.72

4-CAT ENCODING SLOWS DOWN QUANTUM INFORMATION DECAY

Qubit after

Fidelity = 0.26

slide courtesy of R. Schoelkopf and A. Petrenko

QEC = Quantum Error Correction from Parity Monitoring using Field Programmable Gate Array

Ofek & Petrenko et al., Nature (2016)

MULTI-CAVITY ARCHITECTURES

The “Y-mon”!!other experiments currently

underway in the lab with

5 to 10 qubits and 5 to 10 cavities

M. Reagor et al., APL (2013),

C. Axline et al. (in preparation)

Wang and Gao et al., Science (2016)

JOINT WIGNER FUNCTION OF THE TWO-MODE CAT

y- = aA

aB

- -aA

-aB

WJ (bA,bB ): a function in a 4D space

(a =1.92)

Predicted Wigner for ideal state:

P. Millman et al. Eur. Phys. J. D 32, 233 (2005)

Quantum

coherence

Photon

distribution

States live in Hilbert space of

dimension ~ 150!

Ideal:

Measured:

Wang and Gao et al., Science (2016)

SUMMARY AND PERSPECTIVES

• Quantum information protection can be static and/or dynamic

• Cat code encodes information in dynamical states: hardware

efficiency

• Demonstration of operation while protecting

• Attained break-even point in quantum error correction

• Entanglement of two logical qubits

• Combine protection of cat energy loss and parity monitoring

• Demonstrate hierarchical layers of quantum error correction

MEET THE TEAM!

R. Schoelkopf L. Frunzio M. Mirrahimi L. Jiang S. GirvinL. Glazman

A. Narla

K. Serniak

Y. LiuE. Zalys-Geller

M. Hatridge

S. Shankar

M. HaysK. Sliwa Z. Leghtas

A. Roy

U. VoolA. Kou

C. SmithS. TouzardZ. Minev

S. Mundhada

N. FrattiniV. Albert

A. GrimmV. Sivak

protecting quantum superpositions in josephson …

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