Josephson qubits

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 architecture4) 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

III.1) Two-qubit gates

Coupling strategies

2) Tunable coupling

intH (λ)

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

( ) OFFt

Coupling OFF(OFF)

Coupling activatedfor t by ON

( ) ONt

Entangled qubitsInteraction OFF (OFF)

( ) OFFt

III.1) Two-qubit gates

Coupling strategies

2) Tunable coupling

intH (λ)

Entanglement on-demand ???B) Modulate coupling

Coupling OFF(OFF)

( ) OFFt

Coupling ONby modulating

1 2cos t

Coupling OFF(OFF)

III.1) Two-qubit gates

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

F

F

FI FII

(note : idemfor phase qubits)

01, ,

( )2

II

q I z IH F

01, ,

( )2

IIII

q II z IIH F

, ,

( )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

NCC

N

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 00 cos( ) sin( ) 0

( ) 0 sin( ) cos( ) 00 0 0 1

gt i gtU t

i gt gt

00 10 01 11

On resonance, 01 01I II ( )

III.1) Two-qubit gates

i(t)

1 mm

200 µm

qubitsreadout resonator

couplingcapacitor

50 µ

m 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

III.1) Two-qubit gates

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

5

6

7

8

frequ

enci

es (

GH

z)

fI,II/f0

n01I

n01II

ncI

ncII

fI/f00.376

5.14

5.10

5.12

5.16

0.379

2g/ = 9 MHz

Spectroscopy

A. Dewes et al., in preparation

III.1) Two-qubit gates

SWAP between two transmon qubits

5.13 GHz

5.32GHz

6.82 GHz

6.42 GHz

DriveQB I

QB II

QB IIQB I

f01Swap Duration

6.67 GHz

6.03GHz

X

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

III.1) Two-qubit gates

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 IIQB I

f01Swap Duration

6.67 GHz

6.03GHz

X

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

III.1) Two-qubit gates

How to quantify entanglement ??

Need to measure rexp Quantum state tomography

|0>

|1>

X

Z

Y 1 / 2switch yP

/2(X)

III.1) Two-qubit gates

How to quantify entanglement ??

Need to measure rexp Quantum state tomography

|0>

|1>

X

Z

Y 1 / 2switch xP

/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

How to quantify entanglement ??

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 r r r

III.1) Two-qubit gates

0 100 200 300 4000,0

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1,0

|00>

|01>

|10>

|11>

measuredideal

|00>

|11>

|10> |01>

F=98% F=94%

swap duration (ns)

switc

hing

pro

babi

lity

How to quantify entanglement ??

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

SWAP gate of capacitively coupled phase qubitsM. Steffen et al., Science 313, 1423 (2006)

F=0.87

III.1) Two-qubit gates

The Control-Phase gate

Another universal quantum gate : Control-Phase

1 0 0 00 1 0 00 0 1 00 0 0 1

U

00 1001 11

00

10

01

11

Surprisingly, also quite natural with superconducting circuitsthanks 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

III.1) Two-qubit gates

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

Preparation1-qubit rotationsMeasurement

cavity I

RV

Rcos(2 )f t

Rf

x

y

z

Flux bias on right transmon (a.u.)

One-qubit gates: X and Y rotations

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

Preparation1-qubit rotationsMeasurement

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%

One-qubit gates: X and Y rotations

III.1) Two-qubit gates

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

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

0211

Two-excitation manifold

Two-excitation manifold of system

• Avoided crossing (160 MHz)

11 20

Flux bias on right transmon (a.u.)

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

Flux bias on right transmon (a.u.)

11 1e1 11 i

01 e01 01i

10 0e1 10 i

0

2 ( )ft

a at

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 00 1 0 00 0 1 00 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 00 0 00 0 00 0 0

i

i

i

ee

e

U

00 1001 11

00

10

01

11

Implementing C-Phase

11C-Phase11

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

Position: I II III0

“Find the queen!”

Implementing Grover’s search algorithm

“Find x0!”

0

0

0,( )

1,x

f xx 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,x

f xx x

x

“Find the queen!”

Implementing Grover’s search algorithm

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

Position: I II III0

“Find x0!”

0

0

0,( )

1,x

f xx x

x

“Find the queen!”

Implementing Grover’s search algorithm

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

Position: I II III0

“Find x0!”

0

0

0,( )

1,x

f xx x

x

“Find the queen!”

Implementing Grover’s search algorithm

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

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

“Find the queen!”

Implementing Grover’s search algorithm

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

Grover’s algorithm“unknown”unitary

operation:

Challenge:Find the location

of the -1 !!!(= queen)

1 0 0 00 1 0 0ˆ0 0 00 0 1

10

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)

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

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)

(Courtesy Leo DiCarlo)

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

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

(Courtesy Leo DiCarlo)

ideal 0 1 0 10 12

01 1

Apply the “unknown”function, and mark the solution

10

1 0 0 00 1 0 00 0 1 00 0 0 1

cU

/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)

(Courtesy Leo DiCarlo)

Some more 1-qubitrotations…

Now we arrive in one of the four

Bell states

ideal1 112

00

/2yR /2

yR

/2yR

0

010

/2yR

/2yR

/2yR

00

oracle

b c d f

e

g

Grover step-by-step

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

(Courtesy Leo DiCarlo)

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

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

(Courtesy Leo DiCarlo)

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

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

(Courtesy Leo DiCarlo)

III.3) Architecture

Towards a scalable architecture ??

1) Resonator as quantum bus

….

|0>

|register>

III.3) Architecture

Towards a scalable architecture ??

1) Resonator as quantum bus

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

….

|0>

|register>

III.3) Architecture

Towards a scalable architecture ??

1) Resonator as quantum bus

A) Transfer Qi state to resonator2) Control-Phase Gate between any pair of qubits Qi and Qj

….

SWAP

III.3) Architecture

Towards a scalable architecture ??

1) Resonator as quantum bus

A) Transfer Qi state to resonator2) Control-Phase Gate between any pair of qubits Qi and Qj

….

B) Control-Phase between Qj and resonator

C-Phase

III.3) Architecture

Towards a scalable architecture ??

1) Resonator as quantum bus

A) Transfer Qi state to resonator2) Control-Phase Gate between any pair of qubits Qi and Qj

….

B) Control-Phase between Qj and resonatorC) 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

frequ

ency

q

single gate

memory

coupled gate

measure(tunneling)

dampedresonators

zeroing

(courtesy J. Martinis)

RezQu Operations

Idling

1q 2q

r

22

2

Dg

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

'qi-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=60msT2=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, WANTED