Download IBM Josephson junction qubit

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 (>104 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.
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 (>104 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
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:
PRL (2004)
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-Coherence time up to 4lsec
-Improved long term stability
-Scalable?
“Yale” Josephson junction qubit
Coherence time again c. 0.5 ls (in
Ramsey fringe experiment)
But fringe visibility > 90% !
Nature, 2004
IBM Josephson junction qubit
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“qubit = circulation
of electric current
in one direction or
another (????)
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
-- Up to 90% visibility
-- 40nsec decay
-- reasonable long term
stability
What are they?
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Simple electric circuit…
L
C
harmonic oscillator with resonant
frequency
0  1 / LC
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”
Energy

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F
1. Loop: inductance L, energy 2/L
2. Josephson junction:
critical current Ic,
energy Ic cos 
3. External bias energy
(flux quantization
effect): F/L
Josephson phase 
Textbook (classical) SQUID characteristic:
the “washboard”
Energy

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F
1. Loop: inductance L, energy 2/L
2. Josephson junction:
critical current Ic,
energy Ic cos 
3. External bias energy
(flux quantization
effect): F/L
Josephson phase 
Junction capacitance C, plays role of particle mass
Quantum SQUID characteristic:
the “washboard”
Energy

Quantum energy levels
Josephson phase 
Junction capacitance C, plays role of particle mass
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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.
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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?
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not really…
What’s an inductor (linear
or nonlinear)?
F  LI ,
(instantaneous)
1
IL F
I  L F 
Ideal Josephson junction:
1
F is the magnetic flux
 is the superconducting phase
produced by the
inductor
difference across the barrier
F V
F0
2

(Faraday)

 V
(Josephson’s second law)
F0  h / e
flux quantum
not really…
What’s an inductor (linear
or nonlinear)?
F  LI ,
1
IL F
I  L F 
Ideal Josephson junction:
1
F is the magnetic flux
 is the superconducting phase
produced by the
inductor
difference across the barrier
F V
F0
2

(Faraday)

 V
(Josephson’s second law)
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
--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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Graph formalism
1. 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
e.g.,
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NB: this introduces
submatix of F labeled
by branch type
Circuit equations in the graph formalism:
Kirchhoff’s current laws:
V:
I:
F:
Kirchhoff’s voltage laws:
branch voltages
branch currents
external fluxes threading
loops
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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…
the equation of motion (continued):
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.
the equation of motion (continued):
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The lossless parts of this equation arise from a simple Hamiltonian:
H; U=exp(iHt)
the equation of motion (continued):
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 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:
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IBM Josephson junction qubit
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
x
--treat two transverse directions
(blue) as “fast” coordinates
using Born-Oppenheimer
Veff x   Vline x   12 trans,1  12 trans, 2
Extras
IBM Josephson junction qubit:
features of 1-D potential
Veff x 
well asymmetry
 0
barrier height h
x
IBM Josephson junction qubit:
features of 1-D potential
L
R
Well energy
levels, ignoring
tunnel splitting
IBM Josephson junction qubit:
features of 1-D potential
L
well
energy
levels –
tunnel split
into Symmetric and
Antisymmetric states
IBM Josephson junction qubit:
features of 1-D potential
R
well
energy
levels –
tunnel split
into Symmetric and
Antisymmetric states
IBM Josephson junction qubit:
features of 1-D potential
L
A
S
R
well
energy
levels –
tunnel split
into Symmetric and
Antisymmetric states
IBM Josephson junction qubit:
features of 1-D potential
1
L  R
S 
2

well
energy
levels –
tunnel split
into Symmetric and
Antisymmetric states
IBM Josephson junction qubit:
features of 1-D potential
1
L  R
A 
2

well
energy
levels –
tunnel split
into Symmetric and
Antisymmetric states
IBM Josephson junction qubit:
scheme of operation:
--fix  to be zero
--initialize qubit in state
L 
1
S  A
2

--pulse small loop flux, reducing
barrier height h
well asymmetry
 0
barrier height h
x
IBM Josephson junction qubit:
scheme of operation:
--fix  to be zero
--initialize qubit in state
L 
1
S  A
2
splitting  exp FC 

--pulse small loop flux, reducing
barrier height h
A
energy
splitting
S
control flux FC
IBM Josephson junction qubit:
scheme of operation:
--fix  to be zero
--initialize qubit in state
L 
1
S  A
2
splitting  exp FC 

--pulse small loop flux, reducing
barrier height h
A
energy
splitting
S
control flux FC
IBM Josephson junction qubit:
scheme of operation:
--fix  to be zero
--initialize qubit in state
L 
1
S  A
2
splitting  exp FC 

--pulse small loop flux, reducing
barrier height h
--state acquires phase shift

1
S  ei A
2
A
energy
splitting

--in the original basis, this
corresponds to rotating
between L and R:
cos  L  i sin  R
S
control flux FC
“100% visibility”
IBM Josephson junction qubit:
scheme of operation:
--fix  to be small
--initialize qubit in state
L 
1
S  A
2
splitting  exp FC 

A
--pulse small loop flux, reducing
barrier height h
energy
splitting
S
N.B. –
eigenstates are
control flux FC
L
and
R
The idea of a “portal”:
splitting  exp FC 
A
energy
splitting
R
L
portal
S
control flux FC
and
--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
D exp. increases
in time,
D = D 0exp(t/ ).
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 D exponentially increasing
in time, D = D 0 exp(t/ ),
can be solved exactly … the
spinor wavefunction is
  c t  L  c t  R ,
c  exp t / 2  J 1/ 2i /  D 0 /  exp t /  
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 
IBM Josephson junction qubit
Couple qubit to harmonic oscillator (fundamental mode
of superconducting transmission line). Changes the
energy spectrum to:
IBM Josephson junction qubit
Couple qubit to harmonic oscillator (fundamental mode
of superconducting transmission line). Changes the
energy 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
Good Larmor oscillations
IBM qubit
-- Up to 90% visibility
-- 40nsec decay
-- reasonable long term
stability
What are they?
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Overview:
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1. A “user friendly” procedure: automates the assessment of
different circuit designs
2. Gives some new views of existing circuits and their analysis
3. A “meta-theory” – aids the development of approximate theories
at many levels
4. BUT – it is the “orthodox” theory of decoherence – exotic effects
like nuclear-spin dephasing not captured by this analysis.
Adiabatic Q. C.
1.
2.
3.
4.
5.
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.
1.
2.
3.
4.
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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