Quantum computation with solid state devices

-“Theoretical aspects of superconducting qubits”

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

Rosario Fazio

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

“DiVincenzo list”

(Esteve)

(Averin)

Geometric quantum computation

Applications

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

Solid state qubits

Advantages- Scalability- Flexibility in the design

Disadvantages- Static errors- Environment

Qubit = two state system

How to go from N-dimensional Hilbert space (N >> 1)

to atwo-dimensional one?

All Cooper pairs are ``locked'' into the same quantum state

)()()( ris ernr

There is a gap in the excitation spectrum

T/Tc

Quasi-particle spectrum

•Cooper pairs also tunnel through a tunnel barrier

•a dc current can flow when no voltage is applied

•A small applied voltage results in an alternating current

Josephson junction

t)eV(II J 2sin

sinJII

21

Vet

2 relation Josephson

Energy of the ground state ~ -EJcos

SQUID Loop

ehc20

LR

nRL 220

RLI sinsin

0

2cos

JeffJ EI

Dynamics of a Josephson junction

+ + + + + + +_ _ _ _ _ _ _

HC

Q2

2

cosJE

X

=

0sin2

2

mg

ttm

Mechanical analogy

IIRV

tVC J sin

IItRt

C J

sin1

2

2

JIIU cos)(

U

Washboard potential

Quantum mechanical behaviour

HC

Q2

2

cosJE

The charge and the phase are Canonically conjugated variable

ie

Q

,2

From a many-body wavefunctionto a one (continous) quantum mechanical degree of freedom

Two state system

Josephson qubits

Josephson qubits are realized by a proper embedding of the Josephson junction in a superconducting nanocircuit

Charge qubit

Charge-Phase qubit

Flux qubit

Phase qubit

C

J

EE

1

104

Major difference is in the form of the non-linearity

Phase qubit

Current-biased Josephson junction

)sin()cos()( 0101 tItItIII wswcdcdc

U

2 4 6 8 10 12 14

-8

-6

-4

-2

zH 2

01

z x y

The qubit is manipulated by varying the current

Flux qubit

The qubit is manipulated by varying the flux through the loop f and the potential landscape (by changing EJ)

X

(t)

1 2

Cooper pair box

tunable:tunable: - external (continuous) gate charge nx

- EJ by means of a SQUID loop

JC

x

exVCn xx 2/

Cooper pair box

phase difference

Cooper pair number,

voltage across junction

current through junction

ˆsinˆ

2ˆ

)(2ˆ2

ˆ

1||

||ˆˆ

J

x

i

IdtndeI

nnC

edtd

eV

nne

nnnn

Cooper pair box

Cooper pair box

CHARGE BASIS

Charging Josephson tunneling

nN x nnnn

2JE

nnnnCE 112

IJ

Cj

V

Cx

n

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

Coherent dynamics - experiments

Nakamura et al 1999

0 50 1000

20

40

60

Rab

i fre

quen

cy (M

Hz)

nominal URF

(µV)0.0 0.5 1.0

30

35

40

45

50

55

switc

hing

pro

babi

lity

(%)

RF pulse duration t (s)Vion et al 2002

Chiorescu et al 2003

•Chalmers group•NTT group•…

See also exps bySchoelkopf et al, Yale

NIST

Inductance

EJ2 Cnx Cx

EJ1 C

EJ2 Cnx Cx

EJ1 C

L

Vx

Vx

Charge qubit coupling - 1

yyLEH 2112

Capacitance

EJ2 Cnx Cx

EJ1 C

EJ2 Cnx Cx

EJ1 C

Charge qubit coupling - 2

zzCCEH 2112

Josephson Junction

EJ Cnx Cx

EJ1 C

EJ2 C

Charge qubit coupling - 3

..2 2112 chEH J

Tunable coupling

Untunable couplings = more complicated gating

)2cos(~ qEcc

Variable electrostatic transformer

The coupling can be switched off even in the presence of parasiticcapacitances

The effective coupling is due to the (non-linear) Josephson element

zzvH 2112

Averin & Bruder 03

Leakage

The Hilbert space is larger than the computational spaceConsequences:a) gate operations differ from ideal ones (fidelity)b) the system can leak out from the computational space (leakage)

Leakage One qubit gate Fidelity

Two qubit gate Fidelity

|m+1>|m>

~EcEj

|1>|0>qubit

Sources of decoherence in charge qubits

Z

electromagnetic fluctuations of the circuit (gaussian) discrete noise due to fluctuating

background charges (BC)trapped in the substrate or in the junction

Quasi-particle tunneling

Full density matrix

TRACE OUT the environment

RDM for the qubit: populations and coherences

Reduced dynamics – weak coupling

”Charge degeneracy”

(= 0 , = EJ) no adiabatic term optimal point ”Pure dephasing”

(EJ =0 , = ) no relaxation

Reduced dynamics – weak coupling

Background charges in charge qubits

HQz

x

Fluctuations due to the

environmentE

E is a stray voltage or current or charge polarizing the qubit

electrostatic coupling

Charged switching impurities close to a solid state qubit charged impurities

Electronic bandEdi

+di

1g““Weakly coupled” chargeWeakly coupled” charge

Decoherence only depends on

= oscillator environment

““Strongly coupled” chargeStrongly coupled” charge• large correlation times of environment• discrete nature• keeps memory of initial conditions• saturation effects for g >>1• information beyond needed

/vg

1g

g=v/ weak vs strongly coupled charges

EJ=0 – exact solution

envzz HEH

)()(1010

1111

0000

)0()(

)0()(

)0()(

tiEtet

t

t

Constant of motion

Equbit

E

tiHtiH

HXHH

eett

ln)0()(ln)(

12

12

In the long time behavior for a single Background Charge

1g tv

2

The contribution to dephasing due to “strongly coupled” charges (slow charges) saturates in favour of an almost static energy shift

t

~

1gt 2/vttE t

~

EJ=0 – exact solution

Standard model: BCs distributed according to

with

yield the 1/f power spectrum

from experiments 68 1010 A

Experiments: BCs are responsibe for 1/f noise in SET devices.

Warning: an environment with strong memory effects due to the presence of MANY slow BCs

Background charges and 1/f noise

“Fast” noisein general quantum noise• fast gaussian noise• fast or resonant impurities

)|()]([)()( sltf

tslPtsl

t D

EfH

fzt

slz ˆ

2

1)(

2

1Split

Two-stageelimination

Slow noise ≈ classical noise• slow 1/f noise

Slow vs fast noise

222

• expand to second order in → quadratic noise

• Large Nfl central limit theorem → gaussian distributed 0P

• Slow noise: (t) random adiabatic drive M →adiabatic approximation

• Retain fluctuations of the length of the Hamiltonian → longitudinal noise

Paladino et al. 04

see also Shnirman Makhlin, 04 Rabenstein et al 04

)(/1

ln1641 222

SN M

meas

MC t

dAEv fl

• Static Path Approximation (SPA)

variance

HQ

z

xOptimal point

Initial defocusing due to 1/f noise

Standard measurementsno recalibration

with recalibration

Falci, D’Arrigo, Mastellone, Paladino, PRL 2005, cond-mat/0409522

SPA

HQ

z

xOptimal point

Initial suppressionof the signal due

essentially to inhomogeneuos

broadening(no recalibration)

Initial defocusing due to 1/f noise