Josephson qubits P. Bertet SPEC, CEA Saclay (France), Quantronics group.

Josephson qubits

P. Bertet

SPEC, CEA Saclay (France),Quantronics group

0 100 200 300 4000.0

0.2

0.4

0.6

0.8

1.0

11

00

01

switc

hing

pro

babi

lity

swap duration (ns)

10

Outline

Lecture 1: Basics of superconducting qubits

Lecture 2: Qubit readout and circuit quantum electrodynamics

Lecture 3: 2-qubit gates and quantum processor architectures

1) Two-qubit gates : SWAP gate and Control-Phase gate

2) Two-qubit quantum processor : Grover algorithm

3) Towards a scalable quantum processor architecture

4) Perspectives on superconducting qubits

Requirements for QC

Deterministic, On-DemandEntanglement between Qubits

High-Fidelity Readoutof Individual Qubits

0 1

High-Fidelity Single Qubit Operations

III.1) Two-qubit gates

Coupling strategies

1) Fixed coupling

intH

Entanglement on-demand ???« Tune-and-go » strategy

Coupling effectively OFF

Entangled qubitsInteraction effectively OFF

Coupling activatedin resonance for t

F


Coupling strategies

2) Tunable coupling

intH (λ)

Entanglement on-demand ???A) Tune ON/OFF the coupling with qubits on resonance

( ) OFFt

Coupling OFF(lOFF)

l

Coupling activatedfor t by lON

( ) ONt

Entangled qubitsInteraction OFF (lOFF)

( ) OFFt


Coupling strategies

2) Tunable coupling

intH (λ)

Entanglement on-demand ???B) Modulate coupling

Coupling OFF(lOFF)

( ) OFFt

Coupling ONby modulating l

1 2cos t

Coupling OFF(lOFF)


IN THIS LECTURE : ONLY FIXED COUPLING

How to couple transmon qubits ?1) Direct capacitive coupling

coupling capacitor Cc

Vg,IIVg,I

, , ,

,

,

,

,

,

,

,

ˆˆ( ) (

ˆˆ( ) (

ˆ

) co

ˆ2 (

s

)

s

( )

) co

c II II g II J

c I I g I J I I

c I c III

II II II

g I II g IIcc

I

E EN

E N N E

N N

N

E

E

N

E

H

N

FI FII

(note : idemfor phase qubits)

01, ,

( )

2

II

q I z IH

01, ,

( )

2

IIII

q II z IIH

, ,

( )

c x I x II

I II I II

H g

g

2 ˆ0(2 0) 1ˆ 1I I I II II

cII

II I

g eC

NC

CN

J. Majer et al., Nature 449, 443 (2007)

>>D g

g1 g2

Q I Q IIR

How to couple transmon qubits ?2) Cavity mediated qubit-qubit coupling

geff=g1g2/DQ I Q II

eff eff I II I IIH g III.1) Two-qubit gates

iSWAP Gate

01 01/2 2

I III II I II I IIz zH g

intH

« Natural » universal gate : iSWAP

int

1 0 0 0

0 1/ 2 / 2 0( ) SWAP 2 0 / 2 1/ 2 0

0 0 0 1

iU i

g i

int

1 0 0 0

0 cos( ) sin( ) 0( )

0 sin( ) cos( ) 0

0 0 0 1

gt i gtU t

i gt gt

00 10 01 11

On resonance, 01 01I II ( )


i(t)

1 mm

200 µm

qubitsreadout resonator

couplingcapacitor

50

µm

Josephsonjunction

frequencycontrol

fast flux line

Transmonqubit

λ/4 λ/4JJ

coupling capacitor

Readout Resonator

ie

Example : capacitively coupled transmons with individual readout(Saclay, 2011)

qubit

50 µm

drive &readout

frequency control

Example : capacitively coupled transmons with individual readout


0,0 0,2 0,4 0,6

5

6

7

8

fre

que

ncie

s (G

Hz)

fI,II/f0

n01I

n01II

ncI

ncII

fI/f00.376

5.14

5.10

5.12

5.16

0.379

2g/p = 9 MHz

Spectroscopy

A. Dewes et al., in preparation


SWAP between two transmon qubits

5.13 GHz

5.32GHz

6.82 GHz

6.42 GHz

DriveQB I

QB II

QB II

QB If01

Swap Duration

6.67 GHz

6.03GHz

Xp

0,0

0,2

0,4

0,6

0,8

1,0

11

00

01

10

Swap duration (ns)0 100 200

Psw

itch

(%) Raw data


0 100 200 3000,0

0,2

0,4

0,6

0,8

1,0

swap duration (ns)

SWAP between two transmon qubits

5.13 GHz

5.32GHz

6.82 GHz

6.42 GHz

DriveQB I

QB II

QB II

QB If01

Swap Duration

6.67 GHz

6.03GHz

Xp

Psw

itch

(%) Data

correctedfrom

readouterrors

Swap duration (ns)0 100 200

1001

00

iSWAPIII.1) Two-qubit gates

How to quantify entanglement ??

Need to measure rexp Quantum state tomography

|0>

|1>

X

Z

Y 1 / 2switch zP




|0>

|1>

X

Z

Y 1 / 2switch yP

p/2(X)




|0>

|1>

X

Z

Y 1 / 2switch xP

p/2(Y)

M. Steffen et al., Phys. Rev. Lett. 97, 050502 (2006) III.1) Two-qubit gates

0 20

iSWAPX,Y tomo.

readouts

III

I

II

Z

40 60 80ns

I,X,Y


3*3 rotations*3 independent probabilities (P00,P01,P10) = 27 measured numbers

Fit experimental density matrix rexp

Compute fidelity 1/ 2 1/ 2expth thF Tr


0 100 200 300 4000,0

0,2

0,4

0,6

0,8

1,0

|00>

|01>

|10>

|11>

measuredideal

|00>

|11>

|10> |01>

F=98% F=94%

swap duration (ns)

switc

hing

pro

babi

lity


A. Dewes et al., in preparationIII.1) Two-qubit gates

SWAP gate of capacitively coupled phase qubits

M. Steffen et al., Science 313, 1423 (2006)

F=0.87


The Control-Phase gate

Another universal quantum gate : Control-Phase

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

U

00 1001 11

00

10

01

11

Surprisingly, also quite natural with superconducting circuits

thanks to their multi-level structure

F.W. Strauch et al., PRL 91, 167005 (2003)DiCarlo et al., Nature 460, 240-244 (2009) III.1) Two-qubit gates

Control-Phase with two coupled transmonsDiCarlo et al., Nature 460, 240-244 (2009)

int 1 2/ 1 0 0 1 1 0 21 . .eff L R Leff L R L R RH g h g cc h


Spectroscopy of two qubits + cavity

Qubit-qubit swap interaction

cavity

left qubit

right qubit

Cavity-qubit interactionVacuum Rabi splitting

RVFlux bias on right transmon (a.u.)(Courtesy Leo DiCarlo)III.1) Two-qubit gates

Preparation1-qubit rotationsMeasurement

cavity I

One-qubit gates: X and Y rotations

RV

Lcos(2 )f t

Lf

x

y

z

Flux bias on right transmon (a.u.)(Courtesy Leo DiCarlo)III.1) Two-qubit gates


cavity I

RV

Rcos(2 )f t

Rf

x

y

z

Flux bias on right transmon (a.u.)


(Courtesy Leo DiCarlo)III.1) Two-qubit gates


cavity Q

Rf

RV

Rsin(2 )f t

x

y

z

Flux bias on right transmon (a.u.) seeJ. Chow et al., PRL (2009)

Fidelity = 99%



cavity

Conditionalphase gate

Use control lines to push qubits near a resonance

RV

RVFlux bias on right transmon (a.u.)

Two-qubit gate: turn on interactions


0211

Two-excitation manifold

Two-excitation manifold of system

• Avoided crossing (160 MHz)

11 20




11 1e1 11 i

01 e01 01i

10 0e1 10 i

0

2 ( )ft

a a

t

f t dt

Adiabatic conditional-phase gate

10

01

11

2-excitationmanifold

1-excitationmanifold

0

11 10 01 2 ( )ft

t

t dt

0201 10f f

(Courtesy Leo DiCarlo)

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

U

00 1001 11

00

10

01

11

Adjust timing of flux pulse so that only quantum amplitude of acquires a minus sign:

11

01

10

11

1 0 0 0

0 0 0

0 0 0

0 0 0

i

i

i

e

e

e

U

00 1001 11

00

10

01

11

Implementing C-Phase

11C-Phase11


Position: I II III0

“Find the queen!”

Implementing Grover’s search algorithm

“Find x0!”

0

0

0,( )

1,

xf x

x x

x

DiCarlo et al., Nature 460, 240-244 (2009)

First implementation of q. algorithm with superconducting qubits (using Cphase gate)

(Courtesy Leo DiCarlo)III.2) Two-qubit algorithm

Position: I II III0

“Find x0!”

0

0

0,( )

1,

xf x

x x

x




Position: I II III0

“Find x0!”

0

0

0,( )

1,

xf x

x x

x




Position: I II III0

“Find x0!”

0

0

0,( )

1,

xf x

x x

x




Position: I II III0

Classically, takes on average 2.25 guesses to succeed…

Use QM to “peek” inside all cards, find the queen on first try




Grover’s algorithm“unknown”

unitaryoperation:

Challenge:Find the location

of the -1 !!!(= queen)

1 0 0 0

0 1 0 0ˆ0 0 0

0 0 1

1

0

O

/2yR /2

yR

/2yR

0

0

ij/2

yR

/2yR

/2yR

00

oracle

Previously implemented in NMR: Chuang et al. (1998) Linear optics: Kwiat et al. (2000)

Ion traps: Brickman et al. (2005)


Begin in ground state:

ideal 00

/2yR /2

yR

/2yR

0

010

/2yR

/2yR

/2yR

00b c d f

e

g

oracle

Grover step-by-step

DiCarlo et al., Nature 460, 240 (2009)


Create a maximalsuperposition:look everywhere at once!

ideal 0 1 0 10 12

01

1

/2yR /2

yR

/2yR

0

010

/2yR

/2yR

/2yR

00b c d f

e

g

oracle

Grover step-by-step



ideal 0 1 0 10 12

01

1

Apply the “unknown”function, and mark the solution

10

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

cU

/2yR /2

yR

/2yR

0

010

/2yR

/2yR

/2yR

00b c d f

e

g

oracle

Grover step-by-step



Some more 1-qubitrotations…

Now we arrive in one of the four

Bell states

ideal

111

200

/2yR /2

yR

/2yR

0

010

/2yR

/2yR

/2yR

00

oracle

b c d f

e

g

Grover step-by-step



Another (but known)2-qubit operation now undoes the entanglement and makes an interferencepattern that holds the answer!

ideal 0 1 0 10 12

01

1

/2yR /2

yR

/2yR

0

010

/2yR

/2yR

/2yR

00

oracle

b c d f

e

g

Grover step-by-step



Final 1-qubit rotations reveal theanswer:

The binary representation of “2”!

Fidelity >80%

ideal 10

/2yR /2

yR

/2yR

0

010

/2yR

/2yR

/2yR

00b c d f

e

g

oracle

Grover step-by-step



III.3) Architecture

Towards a scalable architecture ??

1) Resonator as quantum bus

….

|0>

|yregister>

III.3) Architecture



2) Control-Phase Gate between any pair of qubits Qi and Qj

….

|0>

|yregister>

III.3) Architecture



A) Transfer Qi state to resonator


….

SWAP

III.3) Architecture





….

B) Control-Phase between Qj and resonator

C-Phase

III.3) Architecture





….

B) Control-Phase between Qj and resonator

C) Transfer back resonator state to Qi

SWAP

III.3) Architecture

Problems of this architecture

….

1) Off-resonant coupling Qk to resonator Uncontrolled phase errors

III.3) Architecture

Problems of this architecture

….

1) Off-resonant coupling Qk to resonator Uncontrolled phase errors

2) Effective coupling between qubits + spectral crowding

geff geff

RezQu (Resonator + zero Qubit) Architecture

q q q q

memoryresonators

qubits

coupling busresonator

freq

uenc

y

q

single gate

memory

coupled gate

measure(tunneling)

dampedresonators

zeroing

(courtesy J. Martinis)

RezQu Operations

Idling

1q 2q

r

2

2

2

g

Transfers& singlequbit gate

• |g reduces off coupling (>4th order)• Store in resonator (maximum coherence)

q

g

0

0

q

0

0

'q

0

0

g

'q

i-SWAPi-SWAP

C-Z(CNOT class)

q

g

r

'q

0

'r

110211 i

Measure

q

g

tunnele

time

10

21

• Intrinsic transfer 99.9999%

|g |g

(courtesy J. Martinis)

Perspectives on superconducting qubits

1) Better qubits ?? Transmon in a 3D cavity

H. Paik et al., arxiv:quant-ph (2011)

REPRODUCIBLE improvement of coherence time (5 samples)

T1=60ms

T2=15ms

Perspectives on superconducting qubits

Quantum feedback : retroacting on the qubit to stabilize a given quantum state

« Non-linear » Circuit QEDResonator made non-linear by incorporating JJ.Parametric amplification, squeezing, back-action on qubit

Quantum information processing : better gates and more qubits !

Hybrid circuits

CPW resonators : versatile playground for coupling many systems :Electron spins, nanomechanical resonators, cold atoms,Rydberg atoms, qubits, …

PhDs, Postdocs, W

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Josephson qubits P. Bertet SPEC, CEA Saclay (France), Quantronics group.

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