Five criteria for physical implementation of a quantum computer

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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 decoherence time (gt104 operation time)
4. Universal set of gate operations
5. 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.
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Five criteria for physical implementation of a
quantum computer quantum communications

1. Well defined extendible qubit array -stable
   memory
2. Preparable in the 000 state
3. Long decoherence time (gt104 operation time)
4. Universal set of gate operations
5. Single-quantum measurements
6. Interconvert stationary and flying qubits
7. Transmit flying qubits from place to place

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Quantum-dot array proposal
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Josephson junction qubit -- Saclay
Oscillations show rotation of qubit at constant
rate, with noise. Wheres the qubit?
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Delft qubit
PRL (2004)

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-Coherence time up to 4lsec -Improved long term
stability -Scalable?
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Yale Josephson junction qubit
Nature, 2004
Coherence time again c. 0.5 ls (in Ramsey fringe
experiment) But fringe visibility gt 90 !
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IBM Josephson junction qubit
qubit circulation of electric current in one
direction or another (????)
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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
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Good Larmor oscillationsIBM qubit

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-- Up to 90 visibility -- 40nsec decay --
reasonable long term stability What are they?
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Simple electric circuit

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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.
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Textbook (classical) SQUID characteristic the
washboard

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Energy
w
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
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Textbook (classical) SQUID characteristic the
washboard

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Energy
w
Energy
w
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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Quantum SQUID characteristic the washboard

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Energy
w
Quantum energy levels
w
Josephson phase
Junction capacitance C, plays role of particle
mass
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But we will need to learn to deal with

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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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Josephson junction circuits

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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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not really

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Whats 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
(Josephsons second law)
(Faraday)
flux quantum
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not really

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Whats an inductor (linear or nonlinear)?
Ideal Josephson junction
is the magnetic flux produced by the inductor
is the superconducting phase difference across
the barrier
(Josephsons second law)
(Faraday)
Phenomenologically, Josephson junctions are
non-linear inductors.
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So, we now do the systematic quantum theory

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Strategy correspondence principle

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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/Lagrangi
an 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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Graph formalism

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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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graph formalism, continued

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NB this introduces submatix of F labeled by
branch type
e.g.,
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Circuit equations in the graph formalism

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Kirchhoffs current laws
V branch voltages I branch
currents F external fluxes threading
loops
Kirchhoffs voltage laws
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With all this, the equation of motion

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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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the equation of motion (continued)

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All are complicated but straightforward functions
of the topology (F matrices) and the inductance
matrix
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Analysis quantum circuit theory tool

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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 equation of motion (continued)

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The lossless parts of this equation arise from a
simple Hamiltonian
H Uexp(iHt)
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the equation of motion (continued)

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The lossy parts of this equation arise from a
bath Hamiltonian, Via a Caldeira-Leggett
treatment
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Connecting Cadeira Leggett to circuit theory

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Overview of what weve accomplished

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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 -gt Bloch Redfield
  theory
• golden rule (decay rates)
• leakage rates

For example
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IBM Josephson junction qubit
Results for quantum potential of the gradiometer
qubit
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IBM Josephson junction qubitpotential landscape
--Double minimum evident (red streak) --Third
direction very stiff
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IBM Josephson junction qubiteffective 1-D
potential
x
--treat two transverse directions (blue) as
fast coordinates using Born-Oppenheimer
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Extras
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IBM Josephson junction qubitfeatures of 1-D
potential
well asymmetry
barrier height
x
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IBM Josephson junction qubitfeatures of 1-D
potential
Well energy levels, ignoring tunnel splitting
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IBM Josephson junction qubitfeatures of 1-D
potential
well energy levels tunnel split into Symmetric
and Antisymmetric states
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IBM Josephson junction qubitfeatures of 1-D
potential
well energy levels tunnel split into Symmetric
and Antisymmetric states
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IBM Josephson junction qubitfeatures of 1-D
potential
well energy levels tunnel split into Symmetric
and Antisymmetric states
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IBM Josephson junction qubitfeatures of 1-D
potential
well energy levels tunnel split into Symmetric
and Antisymmetric states
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IBM Josephson junction qubitfeatures of 1-D
potential
well energy levels tunnel split into Symmetric
and Antisymmetric states
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IBM Josephson junction qubitscheme of
operation
--fix e to be zero --initialize qubit in
state --pulse small loop flux, reducing
barrier height h
well asymmetry
barrier height
x
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IBM Josephson junction qubitscheme of
operation
--fix e to be zero --initialize qubit in
state --pulse small loop flux, reducing
barrier height h
energy splitting
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IBM Josephson junction qubitscheme of
operation
--fix e to be zero --initialize qubit in
state --pulse small loop flux, reducing
barrier height h
energy splitting
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IBM Josephson junction qubitscheme 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
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IBM Josephson junction qubitscheme 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
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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 D 0exp(t/ t).
and
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IBM Josephson junction qubitanalyzing 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 D 0
exp(t/ t), can be solved exactly the
spinor wavefunction is
Which means that the visibility is high so long as
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Problem

• Tunnel splitting exponentially sensitive to
  control flux
• Flux noise will seriously impair visiblity
• Solution ?

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IBM Josephson junction qubit
Couple qubit to harmonic oscillator (fundamental
mode of superconducting transmission line).
Changes the energy spectrum to
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IBM Josephson junction qubit
Couple qubit to harmonic oscillator (fundamental
mode of superconducting transmission line).
Changes the energy spectrum to
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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
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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
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Good Larmor oscillationsIBM qubit

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-- Up to 90 visibility -- 40nsec decay --
reasonable long term stability What are they?
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Overview

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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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Adiabatic Q. C.

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1. Farhi et al idea
2. Feynmann 84 wavepacket propagation idea
3. Aharonov et al connection to adiabatic Q. C.
4. 4-locality, 2-locality effective Hamiltonians
5. Problem polynomial gap

Topological Q. C.

1. Kitaev toric code
2. Kitaev anyons even more complex Hamiltonian
3. Universality honeycomb lattice with field
4. Fractional quantum Hall states 5/2, 13/5