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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 (>104 operation time)

Universal set of gate operations

Single-quantum measurements

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.

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

Science 296, 886 (2002)

Oscillations show rotation of qubit at

constant rate, with noise.

Where’s the qubit?

PRL (2004)

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-Coherence time up to 4lsec

-Improved long term stability

-Scalable?

Nature, 2004

Coherence time again c. 0.5 ls (in

Ramsey fringe experiment)

But fringe visibility > 90% !

IBM Josephson junction qubit

“qubit = circulation

of electric current

in one direction or

another (????)

“qubit = circulation

of electric current

in one direction or

another (xxxx)

Understanding systematically the quantum description of

such an electric circuit…

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-- Up to 90% visibility

-- 40nsec decay

-- reasonable long term

stability

What are they?

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L

C

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.

small

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

small

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

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w

Energy

Quantum energy levels

w

Josephson phase

Junction capacitance C, plays role of particle mass

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--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.

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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?

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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

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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.

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--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.

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- 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

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NB: this introduces

submatix of F labeled

by branch type

e.g.,

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Kirchhoff’s current laws:

V: branch voltages

I: branch currents

F: external fluxes threading

loops

Kirchhoff’s voltage laws:

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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…

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All are complicated but straightforward functions of

the topology (F matrices) and the inductance matrix

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Burkard, Koch, DiVincenzo,

PRB (2004).

Conclusion from this analysis: 50-ohm

Johnson noise not limiting coherence time.

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

H; U=exp(iHt)

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

Via a Caldeira-Leggett treatment:

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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:

Results for quantum potential of the gradiometer qubit…

--Double minimum evident

(red streak)

--Third direction very “stiff”

x

--treat two transverse directions

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

well asymmetry

barrier height

x

Well energy

levels, ignoring

tunnel splitting

well

energy

levels –

tunnel split

into Symmetric and

Antisymmetric states

well

energy

levels –

tunnel split

into Symmetric and

Antisymmetric states

well

energy

levels –

tunnel split

into Symmetric and

Antisymmetric states

well

energy

levels –

tunnel split

into Symmetric and

Antisymmetric states

well

energy

levels –

tunnel split

into Symmetric and

Antisymmetric states

--fix e to be zero

--initialize qubit in state

--pulse small loop flux, reducing

barrier height h

well asymmetry

barrier height

x

--fix e to be zero

--initialize qubit in state

--pulse small loop flux, reducing

barrier height h

energy

splitting

--fix e to be zero

--initialize qubit in state

--pulse small loop flux, reducing

barrier height h

energy

splitting

--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”

--fix e to be small

--initialize qubit in state

--pulse small loop flux, reducing

barrier height h

energy

splitting

N.B. –

eigenstates are

and

--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

--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

Tunnel splitting exponentially sensitive to control flux

Flux noise will seriously impair visiblity

Solution

Couple qubit to harmonic oscillator (fundamental mode

of superconducting transmission line). Changes the

energy spectrum to:

Couple qubit to harmonic oscillator (fundamental mode

of superconducting transmission line). Changes the

energy spectrum to:

--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

--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

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-- Up to 90% visibility

-- 40nsec decay

-- reasonable long term

stability

What are they?

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- 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.

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- 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