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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 (>10 4 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 (>10 4 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 Science 296, 886 (2002) Oscillations show rotation of qubit at constant rate, with noise. Where’s the qubit?
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Delft qubit: small -Coherence time up to 4 l sec -Improved long term stability -Scalable? PRL (2004)
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“Yale” Josephson junction qubit Nature, 2004 Coherence time again c. 0.5 l s (in Ramsey fringe experiment) But fringe visibility > 90% !
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IBM Josephson junction qubit 1 “qubit = circulation of electric current in one direction or another (????)
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IBM Josephson junction qubit Understanding systematically the quantum description of such an electric circuit… “qubit = circulation of electric current in one direction or another (xxxx)
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small 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… small L C 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” small Energy Josephson phase w w 1. Loop: inductance L, energy w 2 /L 2. Josephson junction: critical current I c, energy I c cos w 3. External bias energy (flux quantization effect): wF /L F
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Textbook (classical) SQUID characteristic: the “washboard” small Energy Josephson phase w w Junction capacitance C, plays role of particle mass Energy w F 1. Loop: inductance L, energy w 2 /L 2. Josephson junction: critical current I c, energy I c cos w 3. External bias energy (flux quantization effect): wF /L
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Quantum SQUID characteristic: the “washboard” small Energy Josephson phase w w Junction capacitance C, plays role of particle mass Quantum energy levels
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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?
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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 circuit element, right?
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not really… small What’s an inductor (linear or nonlinear)? is the magnetic flux produced by the inductor (Faraday) Ideal Josephson junction: is the superconducting phase difference across the barrier (Josephson’s second law) flux quantum (instantaneous)
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not really… small What’s an inductor (linear or nonlinear)? is the magnetic flux produced by the inductor (Faraday) Ideal Josephson junction: is the superconducting phase difference across the barrier (Josephson’s second law) Phenomenologically, Josephson junctions are non-linear inductors.
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So, we now do the systematic quantum theory small
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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.
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Graph formalism small 1.Identify a “tree” of the graph – maximal subgraph containing all nodes and no loops graph tree Branches not in tree are called “chords”; each chord completes a loop
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graph formalism, continued small e.g., NB: this introduces submatix of F labeled by branch type
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Circuit equations in the graph formalism: small Kirchhoff’s current laws: Kirchhoff’s voltage laws: V: branch voltages I: branch currents F : external fluxes threading loops
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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…
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the equation of motion (continued): small All are complicated but straightforward functions of the topology (F matrices) and the inductance matrix
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Analysis – quantum circuit theory tool small 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): small The lossless parts of this equation arise from a simple Hamiltonian: H; U=exp(iHt)
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the equation of motion (continued): small 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: small
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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:
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IBM Josephson junction qubit Results for quantum potential of the gradiometer qubit…
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IBM Josephson junction qubit: potential landscape --Double minimum evident (red streak) --Third direction very “stiff”
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IBM Josephson junction qubit: effective 1-D potential --treat two transverse directions (blue) as “fast” coordinates using Born-Oppenheimer x
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Extras
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IBM Josephson junction qubit: features of 1-D potential x well asymmetry barrier height
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IBM Josephson junction qubit: features of 1-D potential Well energy levels, ignoring tunnel splitting
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IBM Josephson junction qubit: features of 1-D potential well energy levels – tunnel split into Symmetric and Antisymmetric states
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IBM Josephson junction qubit: features of 1-D potential well energy levels – tunnel split into Symmetric and Antisymmetric states
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IBM Josephson junction qubit: features of 1-D potential well energy levels – tunnel split into Symmetric and Antisymmetric states
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IBM Josephson junction qubit: features of 1-D potential well energy levels – tunnel split into Symmetric and Antisymmetric states
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IBM Josephson junction qubit: features of 1-D potential well energy levels – tunnel split into Symmetric and Antisymmetric states
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IBM Josephson junction qubit: scheme of operation: x well asymmetry barrier height --fix e to be zero --initialize qubit in state --pulse small loop flux, reducing barrier height h
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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
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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
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IBM Josephson junction qubit: scheme of operation: energy splitting --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: “100% visibility”
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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
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The idea of a “portal”: energy splitting --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: where tunnel splitting D exp. increases in time, D = D 0 exp(t/ t ). and portal
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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 = 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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small Good Larmor oscillations IBM qubit -- Up to 90% visibility -- 40nsec decay -- reasonable long term stability What are they?
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Overview: small 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.
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Adiabatic Q. C. small 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
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