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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 (>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) small -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 1 “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? small small 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 small 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 small 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 small 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. small small 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? small not really… What’s an inductor (linear or nonlinear)? F LI , (instantaneous) 1 IL 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 IL 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. small small So, we now do the systematic quantum theory small 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. small 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., small 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 small 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): All are complicated but straightforward functions of the topology (F matrices) and the inductance matrix small Analysis – quantum circuit theory tool small 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): The lossy parts of this equation arise from a bath Hamiltonian, Via a Caldeira-Leggett treatment: small Connecting Cadeira Leggett to circuit theory: small 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: small 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/ 2i / 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? small 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. 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 small