Five criteria for physical implementation of a quantum computer
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Well defined extendible qubit array -stable memory Preparable in the “000…” state Long decoherence time (>10 4 operation time) Universal set of gate operations Single-quantum measurements. Five criteria for physical implementation of a quantum computer.

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Five criteria for physical implementation of a quantum computer

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Five criteria for physical implementation of a quantum computer

Well defined extendible qubit array -stable memory

Preparable in the “000…” state

Long decoherence time (>104 operation time)

Universal set of gate operations

Single-quantum measurements

Five criteria for physical implementation of a quantum computer

D. P. DiVincenzo, in Mesoscopic Electron Transport, eds. Sohn, Kowenhoven, Schoen (Kluwer 1997), p. 657, cond-mat/9612126; “The Physical Implementation of Quantum Computation,” Fort. der Physik 48, 771 (2000), quant-ph/0002077.


Five criteria for physical implementation of a quantum computer quantum communications

Well defined extendible qubit array -stable memory

Preparable in the “000…” state

Long decoherence time (>104 operation time)

Universal set of gate operations

Single-quantum measurements

Interconvert stationary and flying qubits

Transmit flying qubits from place to place

Five criteria for physical implementation of a quantum computer& quantum communications


Quantum dot array proposal

Quantum-dot array proposal


Josephson junction qubit saclay

Science 296, 886 (2002)

Josephson junction qubit -- Saclay

Oscillations show rotation of qubit at

constant rate, with noise.

Where’s the qubit?


Delft qubit

Delft qubit:

PRL (2004)

small

-Coherence time up to 4lsec

-Improved long term stability

-Scalable?


Yale josephson junction qubit

Nature, 2004

“Yale” Josephson junction qubit

Coherence time again c. 0.5 ls (in

Ramsey fringe experiment)

But fringe visibility > 90% !


Five criteria for physical implementation of a quantum computer

IBM Josephson junction qubit

1

“qubit = circulation

of electric current

in one direction or

another (????)


Ibm josephson junction qubit

IBM Josephson junction qubit

“qubit = circulation

of electric current

in one direction or

another (xxxx)

Understanding systematically the quantum description of

such an electric circuit…


Good larmor oscillations ibm qubit

small

Good Larmor oscillationsIBM qubit

-- Up to 90% visibility

-- 40nsec decay

-- reasonable long term

stability

What are they?


Simple electric circuit

small

L

C

Simple electric circuit…

harmonic oscillator with resonant

frequency

Quantum mechanically, like a kind of atom (with harmonic potential):

x is any circuit variable

(capacitor charge/current/voltage,

Inductor flux/current/voltage)

That is to say, it is a

“macroscopic” variable that is

being quantized.


Textbook classical squid characteristic the washboard

small

Textbook (classical) SQUID characteristic: the “washboard”

w

Energy

F

1. Loop: inductance L, energy w2/L

2. Josephson junction:

critical current Ic,

energy Ic cos w

3. External bias energy

(flux quantization

effect): wF/L

w

Josephson phase


Textbook classical squid characteristic the washboard1

small

Textbook (classical) SQUID characteristic: the “washboard”

w

w

Energy

Energy

F

1. Loop: inductance L, energy w2/L

2. Josephson junction:

critical current Ic,

energy Ic cos w

3. External bias energy

(flux quantization

effect): wF/L

w

Josephson phase

Junction capacitance C, plays role of particle mass


Quantum squid characteristic the washboard

small

Quantum SQUID characteristic: the “washboard”

w

Energy

Quantum energy levels

w

Josephson phase

Junction capacitance C, plays role of particle mass


But we will need to learn to deal with

small

But we will need to learn to deal with…

--Josephson junctions

--current sources

--resistances and impedances

--mutual inductances

--non-linear circuit elements?

G. Burkard, R. H. Koch, and D. P. DiVincenzo, “Multi-level quantum description of decoherence in superconducting flux qubits,” Phys. Rev. B 69, 064503 (2004); cond-mat/0308025.


Josephson junction circuits

small

Josephson junction circuits

Practical Josephson junction is a combination of three electrical elements:

Ideal Josephson junction (x in circuit):

current controlled by difference in

superconducting phase phi across the

tunnel junction:

Completely new electrical circuit

element, right?


Not really

small

not really…

What’s an inductor (linear or nonlinear)?

(instantaneous)

Ideal Josephson junction:

is the magnetic flux

produced by the

inductor

is the superconducting phase

difference across the barrier

(Josephson’s second law)

(Faraday)

flux quantum


Not really1

small

not really…

What’s an inductor (linear or nonlinear)?

Ideal Josephson junction:

is the magnetic flux

produced by the

inductor

is the superconducting phase

difference across the barrier

(Josephson’s second law)

(Faraday)

Phenomenologically, Josephson junctions

are non-linear inductors.


So we now do the systematic quantum theory

small

So, we now do the systematic quantum theory


Strategy correspondence principle

small

Strategy: correspondence principle

--Write circuit equations of motion: these are equations of classical

mechanics

--Technical challenge: it is a classical mechanics with constraints;

must find the “unconstrained” set of circuit variables

--find a Hamiltonian/Lagrangian from which these classical

equations of motion arise

--then, quantize!

NB: no BCS theory, no microscopics – this is “phenomenological”,

But based on sound general principles.


Graph formalism

small

Graph formalism

  • Identify a “tree” of the graph – maximal subgraph containing

  • all nodes and no loops

Branches not in tree are called “chords”; each chord completes a loop

tree

graph


Graph formalism continued

small

graph formalism, continued

NB: this introduces

submatix of F labeled

by branch type

e.g.,


Circuit equations in the graph formalism

small

Circuit equations in the graph formalism:

Kirchhoff’s current laws:

V: branch voltages

I: branch currents

F: external fluxes threading

loops

Kirchhoff’s voltage laws:


With all this the equation of motion

small

With all this, the equation of motion:

The tricky part: what are the independent degrees of freedom?

If there are no capacitor-only loops (i.e., every loop has an inductance),

then the independent variables are just the Josephson phases, and the

“capacitor phases” (time integral of the voltage):

“just like” the biassed Josephson junction, except…


The equation of motion continued

small

the equation of motion (continued):

All are complicated but straightforward functions of

the topology (F matrices) and the inductance matrix


Analysis quantum circuit theory tool

small

Analysis – quantum circuit theory tool

Burkard, Koch, DiVincenzo,

PRB (2004).

Conclusion from this analysis: 50-ohm

Johnson noise not limiting coherence time.


The equation of motion continued1

the equation of motion (continued):

small

The lossless parts of this equation arise from a simple Hamiltonian:

H; U=exp(iHt)


The equation of motion continued2

small

the equation of motion (continued):

The lossy parts of this equation arise from a bath Hamiltonian,

Via a Caldeira-Leggett treatment:


Connecting cadeira leggett to circuit theory

small

Connecting Cadeira Leggett to circuit theory:


Overview of what we ve accomplished

small

Overview of what we’ve accomplished:

We have a systematic derivation of a general

system-bath Hamiltonian. From this we can proceed to obtain:

  • system master equation

  • spin-boson approximation (two level)

  • Born-Markov approximation -> Bloch Redfield theory

  • golden rule (decay rates)

  • leakage rates

For example:


Ibm josephson junction qubit1

IBM Josephson junction qubit

Results for quantum potential of the gradiometer qubit…


Ibm josephson junction qubit potential landscape

IBM Josephson junction qubit:potential landscape

--Double minimum evident

(red streak)

--Third direction very “stiff”


Ibm josephson junction qubit effective 1 d potential

IBM Josephson junction qubit:effective 1-D potential

x

--treat two transverse directions

(blue) as “fast” coordinates using Born-Oppenheimer


Extras

Extras


Ibm josephson junction qubit features of 1 d potential

IBM Josephson junction qubit:features of 1-D potential

well asymmetry

barrier height

x


Ibm josephson junction qubit features of 1 d potential1

IBM Josephson junction qubit:features of 1-D potential

Well energy

levels, ignoring

tunnel splitting


Ibm josephson junction qubit features of 1 d potential2

IBM Josephson junction qubit:features of 1-D potential

well

energy

levels –

tunnel split

into Symmetric and

Antisymmetric states


Ibm josephson junction qubit features of 1 d potential3

IBM Josephson junction qubit:features of 1-D potential

well

energy

levels –

tunnel split

into Symmetric and

Antisymmetric states


Ibm josephson junction qubit features of 1 d potential4

IBM Josephson junction qubit:features of 1-D potential

well

energy

levels –

tunnel split

into Symmetric and

Antisymmetric states


Ibm josephson junction qubit features of 1 d potential5

IBM Josephson junction qubit:features of 1-D potential

well

energy

levels –

tunnel split

into Symmetric and

Antisymmetric states


Ibm josephson junction qubit features of 1 d potential6

IBM Josephson junction qubit:features of 1-D potential

well

energy

levels –

tunnel split

into Symmetric and

Antisymmetric states


Ibm josephson junction qubit scheme of operation

IBM Josephson junction qubit:scheme of operation:

--fix e to be zero

--initialize qubit in state

--pulse small loop flux, reducing

barrier height h

well asymmetry

barrier height

x


Ibm josephson junction qubit scheme of operation1

IBM Josephson junction qubit:scheme of operation:

--fix e to be zero

--initialize qubit in state

--pulse small loop flux, reducing

barrier height h

energy

splitting


Ibm josephson junction qubit scheme of operation2

IBM Josephson junction qubit:scheme of operation:

--fix e to be zero

--initialize qubit in state

--pulse small loop flux, reducing

barrier height h

energy

splitting


Ibm josephson junction qubit scheme of operation3

IBM Josephson junction qubit:scheme of operation:

--fix e to be zero

--initialize qubit in state

--pulse small loop flux, reducing

barrier height h

--state acquires phase shift

--in the original basis, this

corresponds to rotating

between L and R:

energy

splitting

“100% visibility”


Ibm josephson junction qubit scheme of operation4

IBM Josephson junction qubit:scheme of operation:

--fix e to be small

--initialize qubit in state

--pulse small loop flux, reducing

barrier height h

energy

splitting

N.B. –

eigenstates are

and


The idea of a portal

The idea of a “portal”:

--portal = place in

parameter space where

dynamics goes from

frozen to fast. It is

crucial that residual

asymmetry e be small

while passing the

portal:

energy

splitting

portal

where tunnel splitting D exp. increases

in time,

D = D0exp(t/ t).

and


Ibm josephson junction qubit analyzing the portal

IBM Josephson junction qubit:analyzing the “portal”

--e cannot be fixed to be exactly zero

--full non-adiabatic time evolution of

Schrodinger equation with fixed e and

tunnel splitting D exponentially increasing

in time, D = D0 exp(t/ t),

can be solved exactly … the

spinor wavefunction is

Which means that the visibility is high so long as


Problem

Tunnel splitting exponentially sensitive to control flux

Flux noise will seriously impair visiblity

Solution 

Problem:


Ibm josephson junction qubit2

IBM Josephson junction qubit

Couple qubit to harmonic oscillator (fundamental mode

of superconducting transmission line). Changes the

energy spectrum to:


Ibm josephson junction qubit3

IBM Josephson junction qubit

Couple qubit to harmonic oscillator (fundamental mode

of superconducting transmission line). Changes the

energy spectrum to:


Five criteria for physical implementation of a quantum computer

s

--horizonal lines in

spectrum: harmonic

oscillator levels (indep.

of control flux)

--pulse of flux to go

adiabatically past

anticrossing at B, then

top of pulse is in

very quiet part of the

spectrum


Five criteria for physical implementation of a quantum computer

s

--horizonal lines in

spectrum: harmonic

oscillator levels (indep.

of control flux)

--pulse of flux to go

adiabatically past

anticrossing at B, then

top of pulse is in

very quiet part of the

spectrum


Good larmor oscillations ibm qubit1

small

Good Larmor oscillationsIBM qubit

-- Up to 90% visibility

-- 40nsec decay

-- reasonable long term

stability

What are they?


Overview

small

Overview:

  • A “user friendly” procedure: automates the assessment of

  • different circuit designs

  • Gives some new views of existing circuits and their analysis

  • A “meta-theory” – aids the development of approximate theories

  • at many levels

  • BUT – it is the “orthodox” theory of decoherence – exotic effects

  • like nuclear-spin dephasing not captured by this analysis.


Adiabatic q c

small

Adiabatic Q. C.

  • Farhi et al idea

  • Feynmann ’84: wavepacket propagation idea

  • Aharonov et al: connection to adiabatic Q. C.

  • 4-locality, 2-locality – effective Hamiltonians

  • Problem – polynomial gap…

Topological Q. C.

  • Kitaev: toric code

  • Kitaev: anyons: even more complex Hamiltonian…

  • Universality: honeycomb lattice with field

  • Fractional quantum Hall states: 5/2, 13/5


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