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

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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

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

(Esteve)

(Averin)

Geometric quantum computation

Applications

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

Advantages

- Scalability

- Flexibility in the design

Disadvantages

- Static errors

- Environment

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

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

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 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

Flux qubit

(t)

j2

j1

The qubit is manipulated

by varying the flux through

the loop f and the potential

landscape (by changing EJ)

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

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+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

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

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

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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