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Journal of the Physical Society of Japan

Local Field Response Method Working as Quasi-Quantum Annealing

Tatsuya Tomaru

Published Mar 1, 2019• 0 citations
Publisher

Abstract

The local field response (LFR) method is a method of searching for the ground state in a similar manner to quantum annealing. However, it operates on a classical machine, and quantum effects are phenomenologically introduced: (1) an average quantum effect is incorporated in the response function of the spins to the effective field through a priori information; (2) the effect of linear superposition is introduced by taking two successive time points into account in the spin response, reproducing 〈σz〉 ≈ 0 at band anticrossing points; (3) spin correlations, as an effect of entanglement, are introduced by resequencing the calculation order of the spins at their response to the effective field. In this report, we review these phenomenological methods and their effects.

DOI
10.7566/jpsj.88.061016

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The local field response (LFR) method is a method of searching for the ground state in a similar manner to quantum annealing. However, it operates on a classical machine, and quantum effects are phenomenologically introduced: (1) an average quantum effect is incorporated in the response function of the spins to the effective field through a priori information; (2) the effect of linear superposition is introduced by taking two successive time points into account in the spin response, reproducing 〈σz〉 ≈ 0 at band anticrossing points; (3) spin correlations, as an effect of entanglement, are introduced by resequencing the calculation order of the spins at their response to the effective field. In this report, we review these phenomenological methods and their effects.
The local field response (LFR) method is a way of searching for the ground state in a similar manner to quantum annealing. However, the LFR method operates on a classical machine, and quantum effects are introduced through a priori information and through phenomenological means reflecting the states during the computations. The LFR method has been treated with a one-body approximation, and therefore, the effect of entanglement has not been sufficiently taken into account. In this report, spin correlations are phenomenologically introduced as one of the effects of entanglement, by which multiple tunneling at anticrossing points is taken into account. As a result, the accuracy of solutions for a 128-bit system increases by 31% compared with that without spin correlations.
The local-field response (LFR) method is a way of searching for the ground state of an Ising spin Hamiltonian. Spins evolve by responding to an effective field consisting of gradually decreasing external transverse fields and spin–spin interactions. The spins are treated as classical variables, but quantum effects are taken into account in the response of the spins. The conventional LFR method takes a quantum effect into account through an a priori determined response function. This report not only uses the response function but also considers quantum effects more directly. (1) The effect of band anticrossing is considered. (2) The interaction term and local term in the effective field are appropriately balanced by renormalizing the local term. (3) The a priori determined response function is modified on the basis of a self-field correction. In addition to these quantum-mechanical improvements, (4) the transverse field is modulated with a saw shape to increase opportunities of finding the global minimum, and all spins are inverted several times in the middle of the process in order to expand the search space. As a result, the false probability, the inaccuracy of the solution, and the average number of steps before reaching the final solution decrease to about one-tenth of the corresponding values for the case without improvements.
A computational method called the local-field response method is proposed, where spins evolve by responding to an effective field consisting of gradually decreasing external fields and spin–spin interactions, similarly to what is carried out in adiabatic quantum computing (AQC). This method is partly quantum–mechanical. That is, spins are treated as classical variables, but the response function of the spins to the effective field is determined a priori by referring to a quantum–mechanical calculation that was carried out for similar problems. This novel response function improves the performance of the ground state being maintained in the time evolution compared with the case without a priori information. The performance is numerically checked in an eight-qubit system by solving random-interaction problems of finding their ground states. The false probability decreases by about half as a result of using a priori information. The operation of this method is classical, but it has a quantum–mechanical advantage through a priori information. This method is practically useful because obtaining a complete quantum system is difficult as it stands.
A linear response theory for open quantum systems is formulated by means of the time-local and time-nonlocal quantum master equations, where a relevant quantum system interacts with a thermal reservoir as well as with an external classical field. A linear response function that characterizes how a relaxation process deviates from its intrinsic process by a weak external field is obtained by extracting the linear terms with respect to the external field from the quantum master equation. It consists of four parts. One represents the linear response of a quantum system when system-reservoir correlation at an initial time and correlation between reservoir states at different times are neglected. The others are correction terms due to these effects. The linear response function is compared with the Kubo formula in the usual linear response theory. To investigate the properties of the linear response of an open quantum system, an exactly solvable model for a stochastic dephasing of a two-level system is examined. Furthermore, the method for deriving the linear response function is applied for calculating two-time correlation functions of open quantum systems. It is shown that the quantum regression theorem is not valid for open quantum systems unless their reduced time evolution is Markovian.
In this review, how quantum annealing correction works is described from the statistical physics point of view. Quantum annealing correction (QAC) is a programmable error correction scheme in quantum annealing. It is designed to suppress errors by imposing energy penalties. There have been various experimental tests on D-wave machines demonstrating that QAC significantly improves the success probabilities of finding ground states of Ising models. We analyze QAC in the ferromagnetic and Hopfield models by using the mean field theory. The results of our analyses show that QAC efficiently suppresses the thermal fluctuations. In addition, it can significantly increase the energy gap during annealing by changing the energy penalties. In some instances, exponentially small energy gaps turn into polynomially small energy gaps. This suggests that QAC can drastically improve the computational efficiency.
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.