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Quantum Computers, Algorithms and Chaos , Varenna 5-15 July 2005. Quantum computation with solid state devices - “Theoretical aspects of superconducting qubits”. Rosario Fazio. Scuola Normale Superiore - Pisa. “DiVincenzo list”. Two-state system Preparation of the state

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

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

Rosario Fazio

Scuola Normale Superiore - Pisa


“DiVincenzo list”

  • Two-state system

  • Preparation of the state

  • Controlled time evolution

  • Low decoherence

  • Read-out

(Esteve)

(Averin)

Geometric quantum computation

Applications


Outline

Lecture 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 a

two-dimensional one?


All Cooper pairs are ``locked'' into the

same quantum state


Quasi-particle spectrum

There is a gap in the

excitation spectrum

D

D

T/Tc


j1 I j2

Josephson junction

  • 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

Energy of the ground state

~ -EJcosj


SQUID Loop

F

jR

jL


X

Dynamics of a Josephson junction

+ + + + + + +

_ _ _ _ _ _ _

j1 j2

=


Mechanical analogy


Washboard potential

U(f)


Quantum mechanical behaviour

The charge and the phase are

Canonically conjugated variable

From a many-body wavefunction

to 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

1

104

Major difference is

in the form of the

non-linearity


U(f)

Phase qubit

Current-biased Josephson junction

The qubit is manipulated

by varying the current


X

Flux qubit

(t)

j2

j1

The qubit is manipulated

by varying the flux through

the loop f and the potential

landscape (by changing EJ)


Cooper pair box

tunable: - external (continuous) gate charge nx

- EJ by means of a SQUID loop


Cooper pair box

Cooper pair number,

phase difference

voltage

across junction

current

through junction


Cooper pair box


Cooper pair box

V

IJ

Cx

Cj

E

E

C

J

2

CHARGE BASIS

n

(

)

(

)

å

å

2

-

-

+

+

+

n

n

n

n

n

n

1

n

1

n

x

n

N

Charging

Josephson tunneling


From the CPB to a spin-1/2

H =

In the |0>, |1>

subspace

Hamiltonian of a spin

In a magnetic field

Magnetic field in the xz plane


Coherent dynamics - experiments

Schoelkopf et al, Yale

NIST

Chiorescu et al 2003

Nakamura et al 1999

See also exps by

  • Chalmers group

  • NTT group

Vion et al 2002


Charge qubit coupling - 1

EJ1 C

F

nx Cx

EJ2 C

F

EJ2 C

Vx

EJ1 C

Vx

Inductance

nx Cx

L


Charge qubit coupling - 2

EJ1 C

F

nx Cx

EJ2 C

F

EJ2 C

Capacitance

EJ1 C

nx Cx


Charge qubit coupling - 3

EJ1 C

F

nx Cx

EJ C

F

EJ2 C

Josephson Junction

F


Tunable coupling

Variable electrostatic transformer

Untunable couplings = more complicated gating

The effective coupling is due to

the (non-linear) Josephson element

The coupling can be switched off

even in the presence of parasitic

capacitances

Averin & Bruder 03


|m>

|m+1>

~Ec

qubit

Ej

|0>

|1>

Leakage

The Hilbert space is larger than the computational space

Consequences:

a) gate operations differ from ideal ones (fidelity)

b) the system can leak out from the computational

space (leakage)

Leakage

Two qubit gate Fidelity

One qubit gate Fidelity


Sources of decoherence in charge qubits

electromagnetic fluctuations

of the circuit (gaussian)

discrete noise due to fluctuating

background charges (BC)

trapped in the substrate or in the junction

Z

Quasi-particle tunneling


Reduced dynamics – weak coupling

Full density matrix

TRACE OUT the environment

RDM for the qubit: populations and coherences


Reduced dynamics – weak coupling

  • q=0 ”Charge degeneracy”

    (e = 0 , W = EJ)

    no adiabatic term

    optimal point

  • q=p/2”Pure dephasing”

    (EJ =0 , W = e)

    no relaxation


E

charged impurities

Electronic band

Fluctuations due to the environment

HQ

E

z

di+di

x

Background charges in charge qubits

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

Charged switching impurities

close to a solid state qubit

electrostatic coupling


g=v/g weak vs strongly coupled charges

“Weakly coupled” charge

Decoherenceonlydepends on

= oscillator environment

  • “Strongly coupled” charge

  • large correlation times of environment

  • discrete nature

  • • keeps memory of initial conditions

  • • saturation effects for g >>1

  • • information beyond needed


EJ=0 – exact solution

Constant of motion


~

EJ=0 – exact solution

In the long time behavior for a single Background Charge

~

The contribution to dephasing due to “strongly coupled” charges

(slow charges) saturates in favour of an almost static energy shift


Background charges and 1/f noise

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

devices.

Standard model: BCs distributed according to

with

yield the 1/fpower spectrum

from experiments

Warning:an environment with strong memory effects due

to the presence of MANY slow BCs


Split

Slow vs fast noise

  • “Fast” noise

  • in general quantum noise

  • fast gaussian noise

  • fast or resonant impurities

  • Slow noise ≈ classical noise

  • slow 1/f noise

Two-stage

elimination


Initial defocusing due to 1/f noise

z

HQ

x

  • Large Nfl central limit theorem → gaussian distributed

Optimal point

s

2

Paladino et al. 04

  • Slow noise: x(t) random adiabatic drivegM <W →adiabatic approximation

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

  • Static Path Approximation (SPA)

variance

  • expand to second order in x→ quadratic noise

see also Shnirman Makhlin, 04

Rabenstein et al 04


Initial defocusing due to 1/f noise

z

HQ

Initial suppression

of the signal due essentially to inhomogeneuos

broadening

(no recalibration)

x

Optimal point

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

with recalibration

Standard measurements

no recalibration

SPA


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