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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 computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

“DiVincenzo list”

  • Two-state system

  • Preparation of the state

  • Controlled time evolution

  • Low decoherence

  • Read-out

(Esteve)

(Averin)

Geometric quantum computation

Applications


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Solid state qubits

Advantages

- Scalability

- Flexibility in the design

Disadvantages

- Static errors

- Environment


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Qubit = two state system

How to go from

N-dimensional Hilbert space (N >> 1)

to a

two-dimensional one?


Quantum computation with solid state devices theoretical aspects of superconducting qubits

All Cooper pairs are ``locked'' into the

same quantum state


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Quasi-particle spectrum

There is a gap in the

excitation spectrum

D

D

T/Tc


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

SQUID Loop

F

jR

jL


Quantum computation with solid state devices theoretical aspects of superconducting qubits

X

Dynamics of a Josephson junction

+ + + + + + +

_ _ _ _ _ _ _

j1 j2

=


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Mechanical analogy


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Washboard potential

U(f)


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

U(f)

Phase qubit

Current-biased Josephson junction

The qubit is manipulated

by varying the current


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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)


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Cooper pair box

tunable: - external (continuous) gate charge nx

- EJ by means of a SQUID loop


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Cooper pair box

Cooper pair number,

phase difference

voltage

across junction

current

through junction


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Cooper pair box


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Charge qubit coupling - 1

EJ1 C

F

nx Cx

EJ2 C

F

EJ2 C

Vx

EJ1 C

Vx

Inductance

nx Cx

L


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Charge qubit coupling - 2

EJ1 C

F

nx Cx

EJ2 C

F

EJ2 C

Capacitance

EJ1 C

nx Cx


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Charge qubit coupling - 3

EJ1 C

F

nx Cx

EJ C

F

EJ2 C

Josephson Junction

F


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

|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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

Reduced dynamics – weak coupling

Full density matrix

TRACE OUT the environment

RDM for the qubit: populations and coherences


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

EJ=0 – exact solution

Constant of motion


Quantum computation with solid state devices theoretical aspects of superconducting qubits

~

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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


Quantum computation with solid state devices theoretical aspects of superconducting qubits

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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