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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

Adiabatic cyclic evolution

The Hamiltonian of a quantum systemdepends on a set of external parameters r

The external parameters are changed in time r(t)

Adiabatic approximation holds

e.g. an external magnetic field B

e.g. the direction of B

If the system is in an eigenstateit will adjust to the instantaneousfield

Parallel Transport e(0) After a cyclic change

of r(t) the vector e(t)does NOT come backto the original direction

The angle depends onThe circuit C on the sphere

r(T)=r(0)

e(T) ≠

Quantum Parallel Transport

Schroedinger’s equation implements phase parallel transport

|)]([| tHdt

di r

)]([)()]([)]([ tntEtntH n rrr

)]([)( )()(tneet tidttEi

nn r

Schroedinger’s equation:

Adiabatic approx:Instantaneous eigenstates

Look for a solution:

Berry phase

)()( tndt

dtni

The geometrical phase change of |> along a closedcircuit r(T)=r(0) is given by

C r dnni rrr )()(

- M.V.Berry 1984

Spin ½ in an external field

B

C

The Berry phase is relatedto the solid angle that Csubtends at the degeneracy

)(2

1/ C

Bσ H

Adiabatic condition B << 1

Aharonov-Anandan phase

Geometric phases are associated to the cyclic evolution of the quantum state (not of the Hamiltonian)

Generalization to non-adiabatic evolutions

Consider a state which evolves according to the Schrödiger equation such that

)0()( ieT Cyclic state

- Y. Aharonov and J. Anandan 1987

Aharonov-Anandan phase

dttdt

dtidttHt

TT

00

)(~)(~)(~)(~

Introducing such that ~ )0(~)(~ T

Dynamical phase Geometrical phase

Evolution does not need to be adiabaticAdiabatic changes of the external parameters are a way to have a cyclic stateIn the adiabatic limit )()(~ tnt

Aharonov-Anandan phase (Example)

zBH The Hamiltonian

|)2/sin(|)2/cos()0(| Initial state

|)2/sin(|)2/cos()(| iBtiBt eet evolves as

The state is cyclic after T=/B

)]cos(1[)(~)(~0

dttdt

dti

T

AA

Experimental observations

Geometrical phases have been observed ina variety of systems

Aharonov-Bohm effect

Quantum transport

Nuclear Magnetic Resonance

Molecular spectra

…

see “Geometric Phases in Physics”, A. Shapere and F. Wilczek Eds

Geometric phases in superconducting nanocircuits

Possible exp systems: Superconducting nanocircuits

Implications:

• “Macroscopic” geometric interference

•Solid state quantum computation

•Quantum pumping

-G. Falci, R. Fazio, G.M. Palma, J. Siewert and V. Vedral 2000-F. Wilhelm and J.E. Mooij 2001-X. Wang and K. Matsumoto 2002-L. Faoro, J. Siewert and R. Fazio 2003-M.S. Choi 2003-A. Blais and A.-M. S. Tremblay 2003-M. Cholascinski 2004

From the CPB to a spin-1/2

Hamiltonian of a spin In a magnetic field

In the |0>, |1>subspace

H =Magnetic field in the xz plane

Asymmetric SQUID

H = Ech (n -nx)2 -EJ ( cos (

0

221

221 cos4)()( JJJJJ EEEEE

021

21 tantan JJ

JJ

EE

EE

EJ2 C

Cx

EJ1 C

Vx

From the SQUID-loop to a spin-1/2

In the {|0>, |1>} subspace

HB = - (1/2) B .

Bx = EJ cos By = EJ sin Bz = Ech (1-2nx)

“Geometric” interference in nanocircuits

HB = - (1/2) B . Bx = EJ cos

By = EJ sin

Bz = Ech (1-2nx)B

C

“ ”

In order to make non-trivial loops in the parameterspace need to have both nx and

Berry phase - How to measure

•Initial state

•Sudden switch to nx=1/2

•Adiabatic loop

|0>

(1/2½)[|+> + |->]

(1/2½)[ei+i|+> +e-i-i|->]

Berry phase - How to measure

•Swap the states

•Adiabatic loop with opposite orientation

•Measure the charge

(1/2½)[ei+i|-> +e-i-i|+>]

(1/2½)[e2i|-> +e-2i|+>]

P(2e)=sin22

Quantum computation

• Two-state system• Preparation of the state• Controlled time evolution• Low decoherence• Read-out

Phase shifts of geometric origin

Intrinsic fault-taulerant for area-preserving errors

- J. Jones et al 2000- P. Zanardi and M. Rasetti 1999

Geometric phase shift

1111

1010

0101

0000

2/)1(

2/)1(

2/)0(

2/)0(

i

i

i

i

e

e

e

e

z- interaction

Spin 1 Spin 2

Controlled phase gate

Geometric phase shift

Two Cooper pair boxes coupled via a capacitance

Hcoupling = - EK z1z2

-G. Falci et al 2000

Non-abelian case

λH

,...t,...,λt,λtλλ μ21

iii ηEηH

0exp

0

ψUE(t)dth

i(T)ψ

T

NdλAPU

C

μ

μ

μ ,...1, exp

η

ληA

μμ

When the state of the system is degenerate over the full course of its evolution, the system need not to return to the original eigenstate, but only to one of the degenerate states.

Control parameters:N degenerate

tηtUt

00 0 nt

Adiabatic assumption:

Holonomic quantum computation

System S, with state space H ,

perform universal QC

Dynamical approach

Geometric approach

able to control a set of parameters on which depend a iso-degenerate

family of Hamiltonian

information is encoded in an N degenerate eigenspace C of a distinguished

Hamiltonian

Universal QC over C obtained by adiabatically driving the control parameters

along suitable loops rooted at

Mλ

λH

0λH

γ0λ

Josephson network for HQC

L. Faoro, J. Siewert and R. Fazio, PRL 2003

L.M.Duan et al, Science 296,886 (2001)

There are four charges states |j> corresponding to the position of the excess Cooper pair on island j

One excess Cooper pair in the four-island set up

Josephson network for HQC

0,0,1,0J0,1,0,0JD LM1

0,0,0,1JJ0,0,1,0J0,1,0,0JJD2

M2

L*M

*LR2

DEGENERATE EIGENSTATES WITH 0-ENERGY EIGENVALUE

1,0,

0

21

DD

JJ RL

control parameters

One-bit operations

1111

iΣeU Rotation around the z-axis

yσiΣeU 1

2 Rotation around the y-axis

Intially we set 0 RL JJ so the eigenstates

correspond to the logical states

21 D,D

10 ,

In order to obtain all single qubit operationsexplicit realizations of :

0110 iiyσ

Adiabatic pumping

LR

V1(t)

V2(t) Open systems – modulation of the phase of the scattering matrix

Closed systems – periodic lifting of the Coulomb Blockade

Charge transport, in absence of an external bias,by changing system parameters

Charge is transferred coherently

Quantization of transferred charge

-P.W. Brouwer 1998-….

-H. Pothier et al 1992

Cooper pair pumping vs geometric phases

Relation between the geometric phase and

Cooper pair pumping

- J.E. Avron et al 2000-A. Bender, Y. Gefen, F. Hekking and G. Schoen 2004

- M. Aunola and J. Toppari 2003-R. Fazio and F. Hekking 2004

Cooper pair pumping vs geometric phases

In order to relate the second term to the AA phase

~||~tAAt i Take the derivative with respect

to the external phase

][2/ AADT eQ

dttdt

dti

T

AA 0

)(~)(~ dttHtT

D 0

)()(

Cooper pair sluice

t

Iright coil

Vg

tt

Ipumped

Can be generalized to pump 2Ne per cycle. (N = 1,2,…?)

A. O. Niskanen, J. P. Pekola, and H. Seppä, (2003).

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