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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
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
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% !

slide7

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
smallGood 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
smallTextbook (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
smallTextbook (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
smallQuantum 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
smallBut 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
smallJosephson 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
smallnot 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
smallnot 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.

strategy correspondence principle
smallStrategy: 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
smallGraph 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
smallgraph formalism, continued

NB: this introduces

submatix of F labeled

by branch type

e.g.,

circuit equations in the graph formalism
smallCircuit 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
smallWith 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
smallthe equation of motion (continued):

All are complicated but straightforward functions of

the topology (F matrices) and the inductance matrix

analysis quantum circuit theory tool
smallAnalysis – 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
smallthe equation of motion (continued):

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

Via a Caldeira-Leggett treatment:

overview of what we ve accomplished
smallOverview 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

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:

slide51
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

slide52
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
smallGood Larmor oscillationsIBM qubit

-- Up to 90% visibility

-- 40nsec decay

-- reasonable long term

stability

What are they?

overview
smallOverview:
  • 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
smallAdiabatic 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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