Quantum annealing speedup over simulated annealing on random Ising chains

Zanca, Tommaso; Santoro, Giuseppe E.

doi:10.1103/physrevb.93.224431

Cited by 19 publications

(23 citation statements)

References 55 publications

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“…The typical "slow-down" that SQA data with P → ∞ tend to show for large τ should also not necessarily be taken to imply that there is no quantum speed-up of any type against classical Simulated Annealing (SA). Indeed, based on theoretical arguments 4 , a coherent-QA is expected to show some improvement in the exponent of the logarithmic scaling, ε res (τ ) ∼ [log(γτ )] −ξ , against competing SA strategies: this improvement has been indeed verified on random Ising chains 39,43,44 and on infinitely connected p-spin Ising ferromagnets 52 . Moreover, quite remarkably, non-convex optimization problems are known 35 in which SQA, with the P → ∞ limit properly taken, is definitely much more efficient than its classical SA counterpart.…”

Section: Discussionmentioning

confidence: 97%

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Dynamics of simulated quantum annealing in random Ising chains

et al. 2019

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Simulated Quantum Annealing (SQA), that is emulating a Quantum Annealing (QA) dynamics on a classical computer by a Quantum Monte Carlo whose parameters are changed during the simulation, is a well established computational strategy to cope with the exponentially large Hilbert space. It has enjoyed some early successes but has also raised more recent criticisms. Here we investigate, on the paradigmatic case of a one-dimensional transverse field Ising chain, two issues related to SQA in its Path-Integral implementation: the question of Monte Carlo vs physical (Schrödinger) dynamics and the issue of the imaginary-time continuum limit to eliminate the Trotter error. We show that, while a proper time-continuum limit is able to restitute the correct Kibble-Zurek scaling of the residual energy εres(τ ) ∼ τ −1/2 for the ordered case − − − τ being the total annealing time -, the presence of disorder leads to a characteristic sampling crisis for a large number of Trotter time-slices, in the low-temperature ordered phase. Such sampling problem, in turn, leads to SQA results which are apparently unrelated to the coherent Schrödinger QA even at intermediate τ .

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Section: Discussionmentioning

confidence: 97%

“…where now σ z iσ z i+1 t = ψ(t)|σ z iσ z i+1 |ψ(t) is the quantum average with the time-evolving state |ψ(t) . It can be calculated through time-dependent Bogoljoubov-de Gennes (BdG) equations 39,44 . The residual energy at the end of the annealing is simply obtained as…”

Section: Model and Methodsmentioning

confidence: 99%

Dynamics of simulated quantum annealing in random Ising chains

et al. 2019

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“…56, as recently applied on the random Ising chain problem in Ref. 57, we might transform a classical master equation into an equivalent imaginary-time Schrödinger problem with an appropriate effective Hamiltonian H σ,σ which effectively "symmetrizes" the rate matrix W σ,σ using DB; from there, one would then proceed to study such equivalent imaginary-time Schrödinger problem using the same "total spin technique" employed above for the quantum case. One might do that, but we will not do it here, for a reason that we briefly explain in Appendix B.…”

Section: A Quantum Annealing (Qa-rt)mentioning

confidence: 99%

Direct comparison of quantum and simulated annealing on a fully connected Ising ferromagnet

et al. 2017

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We compare the performance of quantum annealing (QA, through Schrödinger dynamics) and simulated annealing (SA, through a classical master equation) on the p-spin infinite range ferromagnetic Ising model, by slowly driving the system across its equilibrium, quantum or classical, phase transition. When the phase transition is second-order (p = 2, the familiar two-spin Ising interaction) SA shows a remarkable exponential speed-up over QA. For a first-order phase transition (p ≥ 3, i.e., with multi-spin Ising interactions) , in contrast, the classical annealing dynamics appears to remain stuck in the disordered phase, while we have clear evidence that QA shows a residual energy which decreases towards 0 when the total annealing time τ increases, albeit in a rather slow (logarithmic) fashion. This is one of the rare examples where a limited quantum speedup, a speedup by QA over SA, has been shown to exist by direct solutions of the Schrödinger and master equations in combination with a non-equilibrium Landau-Zener analysis. We also analyse the imaginary-time QA dynamics of the model, finding a 1/τ 2 behaviour for all finite values of p, as predicted by the adiabatic theorem of quantum mechanics. The Grover-search limit p(odd) = ∞ is also discussed.

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“…The density of defects is then suppressed logarithmically with quench time d 1 ln Q 2 t , which is quadratically faster than simulated annealing, where defects scale as d 1 ln Q t [33,34]. The existence of these logarithmic scaling laws implies that one has to run exponentially long annealing times to reduce the residual energy of the final state.…”

Section: Causal Origin Of Topological Defectsmentioning

confidence: 99%

Engineering non-equilibrium quantum phase transitions via causally gapped Hamiltonians

2018

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We introduce a phenomenological theory for many-body control of critical phenomena by engineering causally-induced gaps for quantum Hamiltonian systems. The core mechanisms are controlling information flow within and/or between clusters that are created near a quantum critical point. To this end, we construct inhomogeneous quantum phase transitions via designing spatiotemporal quantum fluctuations. We show how non-equilibrium evolution of disordered quantum systems can create new effective correlation length scales and effective dynamical critical exponents. In particular, we construct a class of causally-induced non-adiabatic quantum annealing transitions for strongly disordered quantum Ising chains leading to exponential suppression of topological defects beyond standard Kibble-Zurek predictions. Using exact numerical techniques for 1D quantum Hamiltonian systems, we demonstrate that our approach exponentially outperforms adiabatic quantum computing. Using strong-disorder renormalization group (SDRG), we demonstrate the universality of inhomogeneous quantum critical dynamics and exhibit the reconstructions of causal zones during SDRG flow. We derive a scaling relation for minimal causal gaps showing they narrow more slowly than any polynomial with increasing size of system, in contrast to stretched exponential scaling in standard adiabatic evolution. Furthermore, we demonstrate similar scaling behavior for random cluster-Ising Hamiltonians with higher order interactions.Controlling non-equilibrium dynamics of quantum many-body systems is one of the main challenges in condensed matter physics and quantum control. Such complex quantum systems have very rich parameter space and unusual dynamical properties that makes them very hard to simulate and control as they are driven through critical regions [1]. The main difficulties arise from the fact that these systems generally contain high degree of disorders and effectively low dimensions such that they are not prone to exact analytical treatment or mean-field approximations. In principle their dynamics can be mapped to the dynamics of spin-glass systems that are driven/quenched by external control fields and could experience various first and second-order quantum phase transitions and Griffiths singularities [2,3]. Quantum dynamics of such complex systems, except trivial cases, would be out-of-equilibrium when they are quenched in any finite time. However, such rich dynamical properties could lead to novel computational resources [4][5][6][7] provided that we obtain sufficient degree of control over their dynamics.Adiabatic quantum computation (AQC) has been developed as a particular paradigm that utilize the continuous-time dynamics of driven many-body quantum systems for solving optimization tasks [4,5]. In this model, the solution of a hard combinatorial optimization problem is encoded in the ground state of an interacting many-body system which can be prepared adiabatically from an initially trivial ground state, provided that time evolution is much longer that ...

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Quantum annealing speedup over simulated annealing on random Ising chains

Cited by 19 publications

References 55 publications

Dynamics of simulated quantum annealing in random Ising chains

Dynamics of simulated quantum annealing in random Ising chains

Direct comparison of quantum and simulated annealing on a fully connected Ising ferromagnet

Engineering non-equilibrium quantum phase transitions via causally gapped Hamiltonians

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