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Optimal working point in dissipative quantum annealing

Luca Arceci, Simone Barbarino, Davide Rossini, and Giuseppe E. Santoro
Phys. Rev. B 98, 064307 – Published 27 August 2018
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  • INTRODUCTION
  • MODEL AND METHODS
  • NUMERICAL RESULTS
  • DISCUSSION AND CONCLUSIONS
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We study the effect of a thermal environment on the quantum annealing dynamics of a transverse-field Ising chain. The environment is modeled as a single Ohmic bath of quantum harmonic oscillators weakly interacting with the total transverse magnetization of the chain in a translationally invariant manner. We show that the density of defects generated at the end of the annealing process displays a minimum as a function of the annealing time, the so-called optimal working point, only in rather special regions of the bath temperature and coupling strength plane. We discuss the relevance of our results for current and future experimental implementations with quantum annealing hardware.

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    • Received 12 April 2018
    • Revised 10 July 2018

    DOI:https://doi.org/10.1103/PhysRevB.98.064307

    ©2018 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Adiabatic quantum optimizationDissipative dynamicsLandau-Zener effectOpen quantum systemsQuantum phase transitionsQuantum quenchSpin dynamics
    Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & TechnologyStatistical Physics & Thermodynamics

    Authors & Affiliations

    Luca Arceci1, Simone Barbarino1, Davide Rossini2, and Giuseppe E. Santoro1,3,4

    • 1SISSA, Via Bonomea 265, I-34136 Trieste, Italy
    • 2Dipartimento di Fisica, Università di Pisa and INFN, Largo Pontecorvo 3, I-56127 Pisa, Italy
    • 3CNR-IOM Democritos National Simulation Center, Via Bonomea 265, I-34136 Trieste, Italy
    • 4International Centre for Theoretical Physics (ICTP), P.O. Box 586, I-34014 Trieste, Italy

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    Issue

    Vol. 98, Iss. 6 — 1 August 2018

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    Images

    • Figure 1
      Figure 1

      Density of defects vs annealing time for a quantum Ising chain weakly coupled to an Ohmic bath, at different bath temperatures T, compared to the ideal coherent evolution (KZ) behavior, ndef(τ)∼τ−1/2. The plot highlights the three distinct behaviours we have found: ndef(τ) can (i) display a global minimum (green triangles), (ii) a local minimum (blue circles), and (iii) converge monotonically toward a large-τ thermal plateau (red squares). Here the system-bath coupling constant is kept fixed at α=10−2.

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    • Figure 2
      Figure 2

      Density of defects vs annealing time τ for (a) α=10−3, (b) α=10−2, for different effective temperatures T, as indicated in the legend. The arrows indicate the direction of increasing temperatures. The trend for high T is of AKZ type, with an emergent OWP. At lower T and/or higher α values, a monotonic trend smoothly appears, with the absence of OWP.

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    • Figure 3
      Figure 3

      Density of defects vs annealing time τ for kBT=J, at different coupling strengths α. Note that, for large enough annealing times, ndef(τ) converges toward the thermal value.

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    • Figure 4
      Figure 4

      (a) Dependence of nopt on T, for various values of α. Each curve defines an upper value Tup(α) at which nopt(Tup)=n∞(Tup) (marked by stars), and a lower Tlow(α) at which the local minimum defining nopt disappears. (b) Phase diagram in the T−α plane with Tup(α) and Tlow(α). A proper OWP only exists for T>Tup(α). The black solid line is a fit of Tup(α) using Eq. (17), with C=2.08 and D=12.3. The red dashed line is a fit with Tlow(α)=cα−bα−αc, where we find αc=5.5·10−4,c=3.58, and b=0.22. The shaded area alludes to the typical range of temperatures of interest for the D-Wave hardware [7, 8], with kBT≃12 mK and J≳80 mK.

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    • Figure 5
      Figure 5

      Density of defects vs time for τ=105,kBT=J and different system-bath coupling strengths. The arrow at tc marks the value of t at which the transverse field crosses the critical value, h(tc)=hc. For α=10−2, where the defects density has fully converged (see Fig. 3), ndef(t) is almost superimposed to the exact instantaneous thermal one computed from Eq. (B4).

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    • Figure 6
      Figure 6

      Test for the additivity assumption Eq. (20) for the formation of defects. We compare ndef(t=τ), calculated with the full Bloch-Redfield evolution in Eq. (9) (continuous curves, filled symbols), to the sum of ndefcoh(τ) plus the purely dissipative evolution contribution ndefdiss(τ) from Eq. (21) (dashed curve, empty symbols).

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    • Figure 7
      Figure 7

      Comparison between our dissipative QA results (points) and sudden quenches from h0=10 to h0=0 followed by thermal relaxation (lines), for α=0.01 and different temperatures. Data obtained in both cases by the QME dynamics described in the paper. Horizontal black lines identify the expected thermal values for each temperature.

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    • Figure 8
      Figure 8

      Thermodynamics of the defect density in the transverse-field Ising chain for kBT=J. The PBC thermodynamics (orange solid lines) is calculated with Eq. (B7), while the OBC thermodynamics (red solid lines and diamonds) corresponds to Eq. (B11). The blue solid circles are PBC-QME relaxation dynamics data for N/2 two-level systems for Tb=2T.

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    • Figure 9
      Figure 9

      Difference between density of defects for (a) PBC and (b) OBC and the expected thermal value at the thermodynamic limit, plotted vs the number of sites N, for h=0.5,1. (a) The scaling is exponential in N. (b) The scaling is polynomial in N; our best fit gives a convergence with 1/N.

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