Research ArticleQUANTUM INFORMATION

Robust quantum optimizer with full connectivity

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Science Advances  07 Apr 2017:
Vol. 3, no. 4, e1602273
DOI: 10.1126/sciadv.1602273
  • Fig. 1 Illustration of the bifurcation-based annealing process in two antiferromagnetically coupled KPOs.

    (A to C) Evolution of the two-dimensional potential landscape with increasing two-photon drive strength ε. The negative curvature is a consequence of the negative sign of the Josephson-induced Kerr nonlinearity. The antiferromagnetic case (J < 0), where the antisymmetric mode ab is softer, and thus has a lower bifurcation threshold, than the symmetric mode a + b, is shown. Consequently, the system evolves from the vacuum Embedded Image at ε = 0 to the two-mode cat state Embedded Image at large ε, as shown in (D), where we plot the relevant part of the eigenspectrum in the rotating frame (see the Supplementary Materials). Photon loss events predominantly take place above threshold and induce transitions between the cat states Embedded Image and Embedded Image, as indicated by the red arrows in (D).

  • Fig. 2 Schematics of the proposed superconducting CVIM.

    It consists of a chain of KPOs shunted by an effective inductor, Leff. Each KPO is realized by a capacitively shunted split Josephson junction threaded by an ac flux bias at twice the resonance frequency. The Ising spin variables are encoded in the quantized oscillation phases (either 0 or π) of the KPOs above threshold. The inductive shunt induces all-to-all coupling between the KPOs. To obtain antiferromagnetic coupling, a large-area Josephson junction can be used as a shunt together with a flux bias of Φe = Φ0/2. Homodyne readout of the oscillator phases is enabled via capacitively coupled transmission lines (see the Supplementary Materials).

  • Fig. 3 Comparison between coherent and dissipative quantum annealing for two antiferromagnetically coupled equal KPOs.

    The quantum trajectory Embedded Image is obtained by numerically solving the stochastic Schrödinger equation with the Hamiltonian (3) and the photon loss rate κ. The fidelities with respect to the vacuum Embedded Image as well as the three states Embedded Image and Embedded Image, are shown. The amplitude α is given by Eq. 5. (A) Without dissipation: κ = 0. The system evolves from the vacuum Embedded Image (dashed blue line) at t = 0 to the even parity cat state Embedded Image (full red line) at t = T. The latter state encodes the ground state of the corresponding antiferromagnetic Ising model (J < 0). The population of the odd photon number parity state Embedded Image remains 0 (dashed purple line). The bifurcation dynamics is clearly visible as a kink of the population of Embedded Image, when the drive strength reaches the threshold value Embedded Image (vertical thin dashed black line). (B) With dissipation: κ = 0.01 MHz. A quantum trajectory with six jumps obtained from a Monte Carlo simulation of the dissipative dynamics is shown. A photon loss event induces a transition between the even and odd photon number parity cat states. However, note that both Embedded Image and Embedded Image correctly encode the antiferromagnetic Ising spin correlations. Also, note the absence of jumps below threshold, where the average photon number (thin magenta line) is close to 0. The parameter values used in both simulations are as follows: Δ = −1 MHz, J = −0.5 MHz, K = 0.7 MHz, T = 400 μs, εMAX = 2.0 MHz, and ε(t) = εMAX(t/T).

  • Fig. 4 Comparison of performance between the CVIM (A and B) and a standard discrete qubit-based QA (C and D).

    (A and C) Success probability of the NPP with set A = {4,5,6,7} as a function of the inverse ramp rate (dε/dt)− 1 and the photon loss rate κ in (A) or dephasing rate γ in (C). (B and D) Corresponding mean number of jump events [photon loss for (B) and dephasing events for (D)]. Although the success probability of the QA drops sharply already after a single (on average) dephasing event [see region within the white contour in (D)], the CVIM still succeeds with probability > 0.5 even when more than one photon has been lost [see region within the white contour in (B)]. Also, the adiabatic ramp rate threshold for the CVIM is substantially higher than that for the QA. Finally, the success probability for the CVIM is typically above that for a random guess, ~ 1/8 = 0.125, in the entire region shown above the adiabatic threshold, whereas the success probability of the QA quickly drops below the random guess value delimited by thin solid black contour lines in (A) and (C). The parameter values for the CVIM simulations in (A) and (B) are as follows: Δ = −1.5 MHz, K = 0.6 MHz, εMAX = 2 MHz, and ε(t) = εMAX(t/T) for T in the range (0, 200) μs. The parameters for the QA simulations in (C) and (D) are as follows: ε(t) = εMAX(1 − t/T) with εMAX = 6 MHz and T in the range (0, 600) μs. The dashed black curves in (A) and (C) indicate the points where κT = 1 and γT = 1. Each point in all figures corresponds to an average over 40 trajectories.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/4/e1602273/DC1

    section S1. Derivation of the circuit Hamiltonian

    section S2. Readout and dissipation

    section S3. Two coupled parametric oscillators

    section S4. Robustness to decoherence

    section S5. Quantum trajectory comparison between CVIM and QA

    section S6. Parameter estimates

    section S7. Typical quantum trajectories for the NPP

    fig. S1. Illustration of a superconducting quantum interference device–based KPO circuit.

    fig. S2. Schematics of a multiply connected circuit with logarithmically extended tunability.

    fig. S3. Highest part of the eigenspectrum of two coupled parametric oscillators in the rotating frame.

    fig. S4. Populations of the seven highest eigenstates as a function of the normalized ramp rate.

    fig. S5. Stability diagram for the symmetric mode when J > 0.

    fig. S6. Logarithmic negativity of the two-mode Gaussian state during annealing.

    fig. S7. Conditional success probability conditioned on the total number of jumps.

    fig. S8. Conditional success probability conditioned on the total number of jumps (N = 6) and conditioned on the number of jumps in a given oscillator.

    fig. S9. Effect of dephasing on the discrete qubit QA.

    fig. S10. Ratio of ac amplitudes as a function of Formula.

    fig. S11. Conditional expectation values of the interoscillator phase correlations for Formula.

    fig. S12. Conditional expectation values of the interoscillator phase correlations for Formula.

    fig. S13. Conditional expectation values of the interoscillator phase correlations for Formula.

    References (5158)

  • Supplementary Materials

    This PDF file includes:

    • section S1. Derivation of the circuit Hamiltonian
    • section S2. Readout and dissipation
    • section S3. Two coupled parametric oscillators
    • section S4. Robustness to decoherence
    • section S5. Quantum trajectory comparison between CVIM and QA
    • section S6. Parameter estimates
    • section S7. Typical quantum trajectories for the NPP
    • fig. S1. Illustration of a superconducting quantum interference device–based KPO
      circuit.
    • fig. S2. Schematics of a multiply connected circuit with logarithmically extended
      tunability.
    • fig. S3. Highest part of the eigenspectrum of two coupled parametric oscillators in
      the rotating frame.
    • fig. S4. Populations of the seven highest eigenstates as a function of the
      normalized ramp rate.
    • fig. S5. Stability diagram for the symmetric mode when J > 0.
    • fig. S6. Logarithmic negativity of the two-mode Gaussian state during annealing.
    • fig. S7. Conditional success probability conditioned on the total number of jumps.
    • fig. S8. Conditional success probability conditioned on the total number of jumps
      (N = 6) and conditioned on the number of jumps in a given oscillator.
    • fig. S9. Effect of dephasing on the discrete qubit QA.
    • fig. S10. Ratio of ac amplitudes as a function of c1 = cos(Φdc1 ) .
    • fig. S11. Conditional expectation values of the interoscillator phase correlations
      for ε=100 .
    • fig. S12. Conditional expectation values of the interoscillator phase correlations
      for ε=15 .
    • fig. S13. Conditional expectation values of the interoscillator phase correlations
      for ε=3.
    • References (52, 53, 56–61)

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