Five criteria for physical implementation of a quantum computer 1.Well defined extendible qubit array -stable memory 2.Preparable in the “000…” state 3.Long

Five criteria for physical implementation of a quantum computer

1. Well defined extendible qubit array -stable memory

2. Preparable in the “000…” state3. Long decoherence time (>104 operation time)4. Universal set of gate operations5. 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.

Five criteria for physical implementation of a quantum computer

& quantum communications

1. Well defined extendible qubit array -stable memory

2. Preparable in the “000…” state3. Long decoherence time (>104 operation time)4. Universal set of gate operations5. Single-quantum measurements6. Interconvert stationary and flying qubits7. Transmit flying qubits from place to place

Quantum-dot array proposal

Josephson junction qubit -- Saclay

Science 296, 886 (2002)

Oscillations show rotation of qubit at constant rate, with noise.

Where’s the qubit?

Delft qubit:

small

-Coherence time up to 4sec-Improved long term stability-Scalable?

PRL (2004)

“Yale” Josephson junction qubit Nature, 2004

Coherence time again c. 0.5 s (in Ramsey fringe experiment)But fringe visibility > 90% !

IBM Josephson junction qubit1

“qubit = circulationof electric currentin one direction oranother (????)

IBM Josephson junction qubit

Understanding systematically the quantum description ofsuch an electric circuit…

“qubit = circulationof electric currentin one direction oranother (xxxx)

small

Good Larmor oscillationsIBM qubit

-- Up to 90% visibility-- 40nsec decay-- reasonable long term stability

What are they?

Simple electric circuit… small

L C

harmonic oscillator with resonantfrequency

LC/10

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 isbeing quantized.

Textbook (classical) SQUID characteristic: the “washboard”

small

Energy

Josephson phase

1. Loop: inductance L, energy 2/L

2. Josephson junction: critical current Ic, energy Ic cos 3. External bias energy(flux quantization effect): /L

Textbook (classical) SQUID characteristic: the “washboard”

small

Energy

Josephson phase

Junction capacitance C, plays role of particle mass

Energy

1. Loop: inductance L, energy 2/L

2. Josephson junction: critical current Ic, energy Ic cos 3. External bias energy(flux quantization effect): /L

Quantum SQUID characteristic: the “washboard”

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Energy

Josephson phase

Junction capacitance C, plays role of particle mass

Quantum energy levels

But we will need to learn to deal with…

small

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 junctions--current sources--resistances and impedances--mutual inductances--non-linear circuit elements?

Josephson junction circuits

small

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 circuitelement, right?

not really…small

What’s an inductor (linear or nonlinear)?

1

1

,

LI

LI

LI

is the magnetic fluxproduced by the inductor

V

(Faraday)

Ideal Josephson junction:

is the superconducting phase difference across the barrier

V

20

(Josephson’s second law)

eh /0 flux quantum

(instantaneous)

not really…small

What’s an inductor (linear or nonlinear)?

1

1

,

LI

LI

LI

is the magnetic fluxproduced by the inductor

V

(Faraday)

Ideal Josephson junction:

is the superconducting phase difference across the barrier

V

20

(Josephson’s second law)

Phenomenologically, Josephson junctions are non-linear inductors.

So, we now do the systematic quantum theory

small

Strategy: correspondence principle

small

--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 formalismsmall

1. Identify a “tree” of the graph – maximal subgraph containing all nodes and no loops

graphtree

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

graph formalism, continuedsmall

e.g.,

NB: this introducessubmatix of F labeledby branch type

Circuit equations in the graph formalism:small

Kirchhoff’s current laws:

Kirchhoff’s voltage laws:

V: branch voltages I: branch currents: external fluxes threading loops

With all this, the equation of motion:small

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

All are complicated but straightforward functions of the topology (F matrices) and the inductance matrix

Analysis – quantum circuit theory toolsmall

Burkard, Koch, DiVincenzo,PRB (2004).

Conclusion from this analysis: 50-ohm Johnson noise not limiting coherence time.

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 (continued):small

The lossy parts of this equation arise from a bath Hamiltonian,Via a Caldeira-Leggett treatment:

Connecting Cadeira Leggett to circuit theory:small

Overview of what we’ve accomplished:small

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…

IBM Josephson junction qubit:potential landscape

--Double minimum evident (red streak)--Third direction very “stiff”

IBM Josephson junction qubit:effective 1-D potential

--treat two transverse directions(blue) as “fast” coordinates using Born-Oppenheimer

2,21

1,21

transtranslineeff xVxV

x

Extras

IBM Josephson junction qubit:features of 1-D potential

xVeff

h

x

well asymmetry

barrier height

0


L Well energylevels, ignoringtunnel splitting

R


L

wellenergylevels –tunnel splitinto Symmetric andAntisymmetric states



R

A


L wellenergylevels –tunnel splitinto Symmetric andAntisymmetric states

R

S


RLS 2

1



RLA 2

1


IBM Josephson junction qubit:scheme of operation:

h

x

well asymmetry

barrier height

0

--fix to be zero--initialize qubit in state

--pulse small loop flux, reducing barrier height h

ASL 2

1




ASL 2

1

Cfluxcontrol

A

S

energysplitting

Csplitting exp




ASL 2

1

Cfluxcontrol

A

S

energysplitting

Csplitting exp


ASL 2

1

Cfluxcontrol

A

S

energysplitting

Csplitting exp

ASL 2

1

AeS i2

1

RiL sincos


--pulse small loop flux, reducing barrier height h--state acquires phase shift

--in the original basis, this corresponds to rotating between L and R:

“100% visibility”


--fix to be small--initialize qubit in state


ASL 2

1

Cfluxcontrol

A

S

energysplitting

Csplitting exp

N.B. –eigenstates are L Rand

The idea of a “portal”:

Cfluxcontrol

A

S

energysplitting

Csplitting exp--portal = place in parameter space where dynamics goes from frozen to fast. It is crucial that residual asymmetry be small while passing the portal:

1/ where tunnel splitting exp. increases in time, = 0exp(t/ ).

L

R

and

portal

IBM Josephson junction qubit:analyzing the “portal”

-- cannot be fixed to be exactly zero--full non-adiabatic time evolution of Schrodinger equation with fixed and tunnel splitting exponentially increasing in time, = 0 exp(t/ ), can be solved exactly … the spinor wavefunction is

/exp/2/exp

,

0/2/1 tJtc

RtcLtc

i

Which means that the visibility is high so long as 1/

Problem:

• Tunnel splitting exponentially sensitive to control flux

• Flux noise will seriously impair visiblity

• Solution


Couple qubit to harmonic oscillator (fundamental modeof superconducting transmission line). Changes theenergy spectrum to:


Couple qubit to harmonic oscillator (fundamental modeof superconducting transmission line). Changes theenergy spectrum to:

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

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

small

Good Larmor oscillationsIBM qubit

-- Up to 90% visibility-- 40nsec decay-- reasonable long term stability

What are they?

Overview:small

1. A “user friendly” procedure: automates the assessment of different circuit designs2. Gives some new views of existing circuits and their analysis3. A “meta-theory” – aids the development of approximate theories at many levels4. BUT – it is the “orthodox” theory of decoherence – exotic effects like nuclear-spin dephasing not captured by this analysis.

Adiabatic Q. C. small

1. Farhi et al idea2. Feynmann ’84: wavepacket propagation idea3. Aharonov et al: connection to adiabatic Q. C.4. 4-locality, 2-locality – effective Hamiltonians5. Problem – polynomial gap…

Topological Q. C.

1. Kitaev: toric code2. Kitaev: anyons: even more complex Hamiltonian…3. Universality: honeycomb lattice with field4. Fractional quantum Hall states: 5/2, 13/5

Documents

Five criteria for physical implementation of a quantum computer 1.Well defined extendible qubit array -stable memory 2.Preparable in the “000…” state 3.Long