Monte Carlo studies of quantum and classical annealing on a double well

Stella, Lorenzo; Santoro, Giuseppe E.; Tosatti, Erio

doi:10.1103/physrevb.73.144302

Cited by 20 publications

(26 citation statements)

References 35 publications

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“…The mechanical action S[x(τ )] = τ ∞ (mẋ 2 (τ 1 ) + V (x(τ 1 )))dτ 1 for the wavefunction is associated with the unstable La- This observation may have dramatic consequences. As shown in [61,62,63], for non-potential forces the Lagrangian manifold develops cusp singularities some distance away from the origin, Fig. 4.5(a).…”

Section: Discussionmentioning

confidence: 82%

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Quantum Monte Carlo tunneling from quantum chemistry to quantum annealing

2017

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Quantum Tunneling is ubiquitous across different fields, from quantum chemical reactions, and magnetic materials to quantum simulators and quantum computers. While simulating the realtime quantum dynamics of tunneling is infeasible for high-dimensional systems, quantum tunneling also shows up in quantum Monte Carlo (QMC) simulations that scale polynomially with system size. Here we extend a recent results obtained for quantum spin models [Phys. Rev. Lett. 117, 180402 (2016)], and study high-dimensional continuos variable models for proton transfer reactions. We demonstrate that QMC simulations efficiently recover ground state tunneling rates due to the existence of an instanton path, which always connects the reactant state with the product. We discuss the implications of our results in the context of quantum chemical reactions and quantum annealing, where quantum tunneling is expected to be a valuable resource for solving combinatorial optimization problems.

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Section: Discussionmentioning

confidence: 82%

“…It is much more straightforward to devise models that require multidimensional tunneling in continuos space, rather than spin models. 46,47 To this end we borrow insights from quantum chemistry, where simplified model for characterizing proton tunneling have been devised. 4, 28,48,49 In particular in Ref.…”

Section: Multidimensional Tunnelingmentioning

confidence: 99%

Quantum Monte Carlo tunneling from quantum chemistry to quantum annealing

2017

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“…We shall show some theoretical bases for these conclusions in this paper. Numerical evidence is found in [9,10,11,14,15,16,17,18,19,20,21,22].…”

Section: Introductionmentioning

confidence: 99%

Mathematical foundation of quantum annealing

Morita

Nishimori

2008

Journal of Mathematical Physics

302

282

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Quantum annealing is a generic name of quantum algorithms to use quantum-mechanical fluctuations to search for the solution of optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundation of quantum annealing. In particular, theorems are presented for convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schrödinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence both for the Schrödinger dynamics and the stochastic dynamics in spite of the essential difference of these two types of dynamics. Also described are the prescriptions to reduce errors in the final approximate solution obtained after a long but finite dynamical evolution of quantum annealing. It is shown there that we can reduce errors significantly by an ingenious choice of annealing schedule (time dependence of the control parameter) without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity using a classical-quantum mapping.

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“…QA seems to do better than SA in many problems [7][8][9][10] but cases are known where the opposite is true, notably Boolean Satisfiability (3-SAT) 18 and 3-spin antiferromagnets on regular graphs 19 . Usually, the comparison is done by looking at classical Monte Carlo (MC) SA against Path-Integral MC (PIMC) QA 4,18,[20][21][22][23] , but that raises issues related to the MC dynamics, especially in QA. One of these issues has to do with the nature of the PIMC-QA dynamics, in principle not directly related to the physical dynamics imposed by the Schrödinger equation, but nevertheless apparently matching very well the experimental tests on the D-Wave hardware 11 , and also showing a correct scaling of gaps in simple tunneling problems 24 .…”

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confidence: 99%

Quantum annealing speedup over simulated annealing on random Ising chains

Zanca

Santoro

2016

Phys. Rev. B

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We show clear evidence of a quadratic speedup of a quantum annealing (QA) Schrödinger dynamics over a Glauber master-equation simulated annealing (SA) for a random Ising model in one dimension, via an equal-footing exact deterministic dynamics of the Jordan-Wigner fermionized problems. This is remarkable, in view of the arguments of Katzgraber et al., PRX 4, 021008 (2014), since SA does not encounter any phase transition, while QA does. We also find a second remarkable result: that a "quantum-inspired" imaginary-time Schrödinger QA provides a further exponential speedup, i.e., an asymptotic residual error decreasing as a power-law τ −µ of the annealing time τ .

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Monte Carlo studies of quantum and classical annealing on a double well

Cited by 20 publications

References 35 publications

Quantum Monte Carlo tunneling from quantum chemistry to quantum annealing

Quantum Monte Carlo tunneling from quantum chemistry to quantum annealing

Mathematical foundation of quantum annealing

Quantum annealing speedup over simulated annealing on random Ising chains

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