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arXiv:2309.02735

Fast Simulated Annealing inspired by Quantum Monte Carlo

K. Murashima

Published Sep 6, 2023• 0 citations
arXivHTMLPDF

Abstract

Quantum Monte Carlo (QMC) is commonly used in simulations for Quantum Annealing (QA), but QMC as a heuristic approach has great difficulty in that it takes much time to find minimum energy. It mainly depends on the existence of a trotter layer derived from Suzuki-Trotter decomposition. In this paper, I propose a new approach to calculate it in short time, although it isn't rigorous mathematically. Its validity and advantageous points are also discussed, in comparison with conventional QMC methods.

DOI
10.48550/arxiv.2309.02735

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•Jun 18, 2007•23 citations
Quantum Monte Carlo (QMC) is commonly used in simulations for Quantum Annealing (QA), but QMC as a heuristic approach has great difficulty in that it takes much time to find minimum energy. It mainly depends on the existence of a trotter layer derived from Suzuki-Trotter decomposition. In this paper, I propose a new approach to calculate it in short time, although it isn't rigorous mathematically. Its validity and advantageous points are also discussed, in comparison with conventional QMC methods.
An Ising spin model interacting with neighboring spins has been eagerly studied along with the progress in Quantum Computing (QC). Quantum Monte Carlo (QMC) is often used in simulation, but it is such a heuristic approach that it has great difficulty in that it takes a long time to solve the problem. It mainly depends on the existence of a trotter layer derived from Suzuki-Trotter decomposition. In my previous work, I proposed a new approach to compute it much faster by changing the order in which all energies over the trotter layer are summed and minimized. In the algorithm, a plural of trotter layers could be treated in parallel and Time to Solution (TTS) could be reduced as well as that of Simulated Annealing (SA), although it requires a lot of computational resources.In this paper, in order to balance them, a new approach is proposed, where the number of inner loops is not changed but the number of the trotter layers is smaller. With such a small trotter loop repeated in several times, the proposed method works as so-called partial optimization and it leads to very fast convergence. Moreover, it doesn’t affect the result whether the updating spins are the same or not during the partial optimization. This fact also indicates that the new method would open another door to parallel computation. These results are verified by solving MaxCut problem, GSET.
Recent theoretical and experimental studies have suggested that quantum Monte Carlo (QMC) simulation can behave similarly to quantum annealing (QA). The theoretical analysis was based on calculating transition rates between local minima, in the large spin limit using WentzelKramers-Brillouin (WKB) approximation, for highly symmetric systems of ferromagnetically coupled qubits. The rate of transition was observed to scale the same in QMC and incoherent quantum tunneling, implying that there might be no quantum advantage of QA compared to QMC other than a prefactor. Quantum annealing is believed to provide quantum advantage through large scale superposition and entanglement and not just incoherent tunneling. Even for incoherent tunneling, the scaling similarity with QMC observed above does not hold in general. Here, we compare incoherent tunneling and QMC escape using perturbation theory, which has much wider validity than WKB approximation. We show that the two do not scale the same way when there are multiple homotopy-inequivalent paths for tunneling. We demonstrate through examples that frustration can generate an exponential number of tunneling paths, which under certain conditions can lead to an exponential advantage for incoherent tunneling over classical QMC escape. We provide analytical and numerical evidence for such an advantage and show that it holds beyond perturbation theory.
Quantum Annealing (QA) has already put into practical use and considered useful for solving many social issues, such as reduction of traffic congestion and delivery optimization. But in QA, when the energy difference between the ground state and the first excited state is small, the transition probability between them increases. Therefore, in order to decrease the transition, QA has to be performed at extremely low temperatures. To simulate the situation, it has the problem in that it takes much time. Many studies are underway to accelerate QA theoretically, one of which is to incorporate non-stoquastic Hamiltonian. On the other hand, I proposed Nested Simulated Annealing (NSA) inspired by Quantum Monte Carlo (QMC). I showed the computational speedup could be achieved dramatically, by considering the idea that the effect of flipping a spin preferentially influenced the spins directly interacting with it. Although NSA was based on such classical concept of causality, the hybrid computation both quantum and classical approach worked well. In this paper, in order to discuss the relationship between NSA and non-stoquastic Hamiltonian like XX-interaction, the spins are treated as continuous variables. I derive the formula to calculate the total energy in both with a problem Hamiltonian and the perturbation Hamiltonian induced by a transverse electromagnetic field. And I will show a clear relationship between local-maxima and the convergence speed when calculating it with the binary spin. More precisely, the annealing path with the smallest local-maxima converges fast, even though it consumes fewer computational resources. Therefore, it is possible to induce an optimal adiabatic phase transition by selecting NSA parameters appropriately. This paper shows an effective method to choose the parameters when simulating QMC on a classical computer.
Abstract Quantum annealing aims to provide a faster method than classical computing for finding the minima of complicated functions, and it has created increasing interest in the relaxation dynamics of quantum spin systems. Moreover, problems in quantum annealing caused by first-order phase transitions can be reduced via appropriate temporal adjustment of control parameters, and in order to do this optimally, it is helpful to predict the evolution of the system at the level of macroscopic observables. Solving the dynamics of quantum ensembles is nontrivial, requiring modeling of both the quantum spin system and its interaction with the environment with which it exchanges energy. An alternative approach to the dynamics of quantum spin systems was proposed about a decade ago. It involves creating stochastic proxy dynamics via the Suzuki-Trotter mapping of the quantum ensemble to a classical one (the quantum Monte Carlo method), and deriving from this new dynamics closed macroscopic equations for macroscopic observables using the dynamical replica method. In this chapter, we give an introduction to this approach, focusing on the ideas and assumptions behind the derivations, and on its potential and limitations.
Abstract Quantum Monte Carlo has found a new application far afield from the solution of the Schrodinger equation for many-body problems. In this paper the authors report a method for solving multidimensional optimization problems that is superior, in many cases, to other methods such as conjugate gradient methods, the simplex method, direction-set methods, genetic methods, and classical simulated annealing. The new approach, aptly titled “quantum annealing,” is closely related to classical annealing, in which a physical system such as a metal cluster is slowly cooled and finds its lowest-energy configuration as the temperature drops to zero. In classical annealing the system follows classical-mechanical dynamics as the temperature drops. In quantum annealing a collection of configurations evolves in the the same way as walkers undergoing diffusion and multiplication in diffusion QMC with the added feature of a slowly dropping value for Planck’s constant h. Classical annealing allows the escape from local minima with the aid of thermal fluctuations. Quantum annealing allows such escape by delocalization and tunneling. Since a QMC walker can multiply in number on entering a low-energy region, a single walker can populate an entire region.