Quantum computation with solid state devices - “Theoretical aspects of superconducting qubits”

Quantum computation with solid state devices

-“Theoretical aspects of superconducting qubits”

Quantum Computers, Algorithms and Chaos, Varenna 5-15 July 2005

Rosario Fazio

OutlineLecture 1

- Quantum effects in Josephson junctions- Josephson qubits (charge, flux and phase)- qubit-qubit coupling- mechanisms of decoherence- Leakage

Lecture 2

- Geometric phases- Geometric quantum computation with Josephson qubits- Errors and decoherence

Lecture 3

- Few qubits applications- Quantum state transfer- Quantum cloning

Quantum information protocols without external control

Choose a given model and use just the time evolution (less flexible but more stable)(less flexible but more stable)

Easier to implement in solid state systems

Implementation of Quantum communication schemes in solid state devices

Josephson arrays in quantum communication

Motivations

Alice

Bob

Quantum channel

Protocols

Cloning

Quantum state transfer

Alice

Bob~ |>

Quantum communications with spin chains

|>=a|0>+b|1>

Quantum channel

Alice

Bob

. . .|> ||0> |> |0>

J J J

Initial state

i

zi

iiii hJH 1

S. Bose (2002), M. Christandl et al (2003), F. Verstraete, M. Martin-Delgado and J.I. Cirac (2003), D. Burgarth and S. Bose (2004), D. Burgarth, V. Giovanetti and Bose (2005), V. Giovannetti and R. Fazio (2005), A. Romito, C. Bruder and R. Fazio (2005), G. De Chiara D. Rossini, S. Montangeroand R. Fazio (2005), …

Quantum communications with spin chains

Alice

Bob

. . . J J J

000...0000||

i

zi

iiii hJH 1

Quantum communications with spin chains

000...0000||

• Initial state

• Time evolution

000...0000||

)(||)()( }1,...1{ tUtUTrt LL

1|sin0|cos|

ie

•Sender at site 1•Receiver at site L

Quantum communications with spin chains

000...0000|

i

zi

iiii hJH 1

000...0000|

Quantum communications with spin chains

Total magnetizationIs a constant of motion

ja j |000...1000|

where

thj

j

000...1...000|| Le iHt ||1

dtF

|)(|

Jt

Quality of the channel – average fidelity

Fidelity ~ L-1/3

L

Quantum communications with spin chains

Perfect communications with spin chains

xiJSetU )(

No cloning theorem

U|a>|0> |a>|a>

U(|a>+|b>) |0> |aa>+|bb> ≠ (|a>+|b>) (|a>+|b>)

Quantum cloning

W. Wootters and W. Zurek (1982)

A quantum copying machine does not exist!

Although perfect cloning is not possible …..

… Imperfect cloning has been considered

Quantum cloning

V. Buzek and M. Hillery (1996), D. Bruβ et al (1998), D. Bruβ, A. Ekert and C. Macchiavello (1998), R. Werner (2000), …

A. Lama-Linares et al (2002), De Martini et al (2004),J. Du et al (2004), …

Central quantity

Fidelity for n m cloning

n states to be copied belonging to a portion of the Hilbert space

the m cloned statesare in in the mixedstate

||, mnF

IndependentIndependent on on

83.021 UCF

Universal ClonerUniversal Cloner

Phase Covariant ClonerPhase Covariant Cloner

Examples

854.021 PCCF Fidelity at

the equator

Ry(/2)

21

31

Quantum circuits

• XY Model

• Heisenberg Model

• Ising Model

•Start from the state to clone

•Wait for a time (independent on the state to be cloned)

•Cloning 1 m

| > =|0>

| > = cos|0>+sinei |1>

G. De Chiara et al. (2004,2005)

Spin network cloning

Fid

elity

Heisenberg

Ideal cloner coincides with the XY model

21

Phase covariant cloner

Best cloner

Heisenberg

XY

m1

m

Fid

elity

n m F Jt

2 3 0.94 81

2 5 0.87 73

2 7 0.81 69

3 5 0.97 584

Cloning from n to m

F

Cloning in the presence of noise

/J

Vg

Vg

Josephson coupling realizes the XY model

Quantum cloning with Josephson qubits

..2 2112 ch

EH J

..2 3113 ch

EH J

Vg

Vg

Josephson couplingrealizes the XY coupling

F

U1U2

Quantum cloning with Josephson qubits

Josephson arrays as

artificial 1D magnets

Charge regime - C. Bruder R, Fazio, G. Schön, PRB 47, 342 (1993) Flux regime - L. Levitov, T.P. Orlando, J.B. Mayer, J.E. Mooij cond-mat 0108266

Quantum communication with JJA

Bose-Hubbard = Quantum Phase Model = XXZ model

prep

arat

ion

measurem

entstate propagation

).(2 1 ch

EhUH

iii

J

i

zii

ij

zjij

zi

C. Bruder et al (1993)

Quantum communication with JJAs

Example - N=3 tF

idel

ity

EJ /EC=0.1

C/C0=10

Fidelity ~ 0.999

|)(| tF

| N0|

Averaged over the initial state

Fmax

N

Fidelity vs N

Fmax

C0/C

Dependence on the electrostatic interaction

V

t = tmax|>~

The current is proportional to the fidelity

Vg

VgVg

Vg

The charge state of the N island as a function of tp

Current correlation <I1(t)IN(t+tp)>

Charge correlation <n1(t)nN(t+tp)>

Fidelity ~ 0.95

EJ/EJ=10%

nx/nx=10%

Imperfections – N=3

t

L. Levitov, T.P. Orlando, J.B. Mayer, J.E. Mooij cond-mat 0108266

x x x x x x

State transfer with flux qubits

Alice

Bob

Quantum channel

Entangled

Entanglement sharing

{ {………………………………………..

singletsinglet

● ● ● ● ● ●

● ● ● ● ● ● time

-1 0 1 2 3 4site #

● ● ● ● ● ●

Singlet propagation

Entanglement

Singlet initial state

Entropy for symmetric sites

Entropy for sites (-6,7)

Fidelity to the initial singlet

JJ arrays can be used in quantum communication

Entanglement sharing Quantum Cloning State transfer

Experiments seems to be possible at present

Conclusions