d-wave systems inc. the quantum computing company tm a.m. zagoskin (d-wave systems and ubc) tunable...

D-Wave Systems Inc.THE QUANTUM COMPUTING COMPANYTM

A.M. Zagoskin (D-Wave Systems and UBC)

Tunable coupling of superconducting qubits

Quantum Mechanics on the Large Scale, Banff,

April 12-17, 2003

A. Blais (Yale University)

A. Maassen van den Brink (D-Wave Systems)

A.Yu. Smirnov (D-Wave Systems)

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Direct coupling of superconducting qubits

Capacitive coupling - charge qubits Capacitive coupling- CBJJ qubits

Pashkin et al., Nature 421 (2003) 823

Blais, Maassen van den Brink, and Zagoskin, PRL 90 (2003) 127901

Johnson et al., PRB 67 (2003) 020509(R)

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Il’ichev et al. (2003)

Inductive coupling - 3JJ qubits

Makhlin, Schön, and Shnirman,

Rev.Mod.Phys. 73 (2001) 357

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Paauw et al. (2002)

Inductive+ coupling - 3JJ qubits

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Inductive+ coupling - 3JJ qubits

Akisato, quant-ph_0303128 (2002)

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Coupling through a resonant tank circuit

ab

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Coupling through virtual states

Coupled charge qubits Coupled phase qubits

Makhlin, Schön, and Shnirman, Rev.Mod.Phys. 73 (2001) 357

jy

iy

ji

xjJxiJ

gJ

g

L

EE

CC

CH

ˆˆ

/

)()(22

0

2

int

jy

iy

osc

jx

ix

ji

ji

Ce

BB

L

MH

ˆˆ

/220

2

int

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Qubit-qubit entanglement in cavity QED

Rauschenbeutel et al., Science 288 (2000) 2024

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Superconducting tanks and qubits

Ilichev et al., cond-mat/0303433 (2003)

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Superconducting tanks and qubits

7000~;5.2 QsRabi

Ilichev et al., cond-mat/0303433 (2003)

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Tunable inductance

./2

,/)cos1(

,/sin

0

c

c

c

LI

II

II

Assuming 1, we obtain

)/cos(1)/)(cos(1 cbceff II

L

ItI

LL

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Mediated coupling


Plastina and Falci, cond-mat/0206586 (2002)

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Tuning a large Josephson junction

4/122/1

12

c

b

J

cJ I

I

C

Ie

4

0

)/(1/

J

cbJcb

IIII

c

b

I

I

0JJ

Weak sensitivity to bias noise.

Fine tuning.

Moderate response to bias

current.

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Coupled CBJJs


Qubit

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CBJJ qubit coupled to a tunable bus

.~2cos~

22cos~

20

20

2

c

bqbbbJb

Jb

b

qqqJq

Jq

q

C

QQIE

C

QIE

C

QH

02/

2/0

2/

2/

10

10)2(

qb

bq

EE

EEH

cJ

cJbJq

c

JJbJq CC

C

C

C

~

~

In the {|0q1b>, |1q0b>}-subspace

Coupling parameter

(in resonance)

2,0

2 )/(1/2 cJcJ IICI Josephson frequency

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Coherent qubit-bus oscillations

If Cj=6 pF, Cc=25 fF, Ic=21 A, Ibias=20.8 A,

there are three levels in each well (interlevel spacing ~1 GHz),

and coherent oscillations have period T=h/~40 ns.

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Two-qubit operations:

1000

02

1

2

10

02

1

2

10

0001

SWAP

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Two-qubit operaions:

42;22 21

tt

1t 1t2t

Ib

Ib,2

Ib,1Ib,1

Ib,decouple

"" SWAP

t

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Decoupling of CBJJ qubit from the bus

Decoupling is achieved if decrease the bus current to

Ibias=20.43 A.

The “degenerate” eigenstates are

0.999|0q1b>+0.007|1q0b>+o(10-3)

0.007|0q1b>+0.999|1q0b>+o(10-3)

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Additional noise in CBJJ qubit-bus system

zzbzqxxbxq

zJb

Jb

Jq

JqzJbJqSB

XXXX

H

ˆˆ

ˆ4

ˆ2

;2

122

,

II

II

cJ

J

Noise source: bias current fluctuations:

TkJX

TkRI

BB 2coth)(;

2coth 22

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Additional noise in CBJJ qubit-bus system

TkJ

TT

TkJT

BZ

BX 2

coth)(lim2

1

2;

2coth)/(

2

10

111

21

1

In resonance,

For T = 25 mK, R = 560 k, CJ = 6 pF, Cc = 25 fF, Ic=20 A

T1,2 ~ 1 ms

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Quality of coupled CBJJ qubits


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Coupled “quantroniums”


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Phase qubits coupled through a resonator

Smirnov and Zagoskin, cond-mat/0207214 (2002)

CTIT

LT

Ib Ic CJ

M1M2

I1 I2

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Phase qubits coupled through a resonator

Smirnov and Zagoskin/cond-mat/0207214 (2002)

i

iziT

izi

ixi aaaaH ˆ2/1ˆˆ

2

1

TiiiT

Tiii LLkM

LMI ;

2

If both qubits are in a degeneracy point and near resonance:

i

titii

RWA TiTi easeastH //int ˆˆ

2

1)(

11

11ˆ

s

i

iJC asastH ˆˆ2

1)(

Jaynes-Cummings Hamiltonian

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Qubit + tank

.21int

jnjnjH

0;1;12

1)2,1( nnn

)2,1()2,1(nTn nnE

.0;;1;12

1nn

jjjn

Tn 1

2

t

it

qTqTqT

sin0;1cos1;02

1

2

11;0 )2(

0)1(

0

1n

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Qubit + tank

For a 3JJ phase qubit with Iq~450 nA, L~25 pH, fT~1 GHz

the frequency of the corresponding coherent oscillations is

f0~0.1 GHz.

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State pump

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Equal qubits, equal couplings

.41int

jnjnjH

.01012

1

2/12/1

1

2

1

;01012

1

2/1

1

2/12

1

22114321)4,3(

22114321)2,1(

nn

n

n

n

nn

n

n

n

nnnnnn

nnnnnn

)4,3()4,3(

)2,1()2,1( 2/12

nTn

nTn

nE

nnE

.10;;01;;00;1;11;1 212121214

1nnnn

jjjn

No “classical” entanglement!

Tn 1

3,4

2

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(Teleportation)1/2

2

10010

2

001

2

10010

.10;0;01;0;00;1

2121)3()3(

212121)2,1()2,1(

2121214

20

E

E

jj

For n=0 there are only three constituent states:

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(Teleportation)1/2

Starting from the state in=|0>(a|11>+b|01>) | 02>,

after time t1/2= 2-1/2 / we reach the state

out=|0> | 01>( -a|12>+b|02>)

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(Teleportation)1/2

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Photon-splitting and qubit-qubit entanglement

If instead start from the state ’in=|1>|01>|02>,

,2

10;001;0)2sin(00;1)2cos()( 2121

21

titt

and after tB = 2-3/2 / the Bell state of two qubits is formed:

.2

10;001;0)( 2121 itB

For the above choice of parameters and Q=1000

the decay time in the tank ~ 1 s, while tB ~ 1 ns

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Phase-charge duality

Plastina and Falci, cond-mat/0206586 (2002)

The previous results apply to capacitively coupled charge qubits.

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Phase-charge vocabulary

.

;ˆˆ

;ˆˆ2

1

2

J

zx

kpT

E

s

C

eCg

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Tunable capacitance

Averin and Bruder, cond-mat/0304166 (2003)

Vg

C1C2

EJC

Effective capacitance can also be tuned

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Conclusions

Mediated coupling of qubits through tunable superconducting

tank circuits allows:

•Two-qubit operations; •Teleportation of qubit state;•Effective coupling/uncoupling of non-tunable qubits; •Coupling of qubits of different types; •Tank protects qubits from the external noise;•Independent optimization of qubit and tank parameters;•Can be realized with available experimental techniques.

d-wave systems inc. the quantum computing company tm a.m. zagoskin (d-wave systems and ubc) tunable...

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