Post on 18-Dec-2015
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
Josephson junction qubit -- Saclay
Science 296, 886 (2002)
Oscillations show rotation of qubit at constant rate, with noise.
Where’s the qubit?
“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”
small
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.
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
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:
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:
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
IBM Josephson junction qubit:features of 1-D potential
L Well energylevels, ignoringtunnel splitting
R
IBM Josephson junction qubit:features of 1-D potential
L
wellenergylevels –tunnel splitinto Symmetric andAntisymmetric states
IBM Josephson junction qubit:features of 1-D potential
wellenergylevels –tunnel splitinto Symmetric andAntisymmetric states
R
A
IBM Josephson junction qubit:features of 1-D potential
L wellenergylevels –tunnel splitinto Symmetric andAntisymmetric states
R
S
IBM Josephson junction qubit:features of 1-D potential
RLS 2
1
wellenergylevels –tunnel splitinto Symmetric andAntisymmetric states
IBM Josephson junction qubit:features of 1-D potential
RLA 2
1
wellenergylevels –tunnel splitinto Symmetric andAntisymmetric states
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
IBM Josephson junction qubit:scheme of operation:
--fix to be zero--initialize qubit in state
--pulse small loop flux, reducing barrier height h
ASL 2
1
Cfluxcontrol
A
S
energysplitting
Csplitting exp
IBM Josephson junction qubit:scheme of operation:
--fix to be zero--initialize qubit in state
--pulse small loop flux, reducing barrier height h
ASL 2
1
Cfluxcontrol
A
S
energysplitting
Csplitting exp
IBM Josephson junction qubit:scheme of operation:
ASL 2
1
Cfluxcontrol
A
S
energysplitting
Csplitting exp
ASL 2
1
AeS i2
1
RiL sincos
--fix 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:
“100% visibility”
IBM Josephson junction qubit:scheme of operation:
--fix to be small--initialize qubit in state
--pulse small loop flux, reducing barrier height h
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
IBM Josephson junction qubit
Couple qubit to harmonic oscillator (fundamental modeof superconducting transmission line). Changes theenergy spectrum to:
IBM Josephson junction qubit
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