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Journal of the Physical Society of Japan

Improved Local Field Response Method Working as Quasi-Quantum Annealing

Tatsuya Tomaru

Published Apr 18, 2017• 2 citations
Publisher

Abstract

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.

DOI
10.7566/jpsj.86.054801

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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.
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.
We study the problem to infer the original ground state of a spin-glass Hamiltonian out of the information from the Hamiltonian with interactions deviated from the original ones. Our motivation comes from quantum annealing on a real device in which the values of interactions are degraded by noise. We show numerically for quasi-one-dimensional systems that the Hamming distance between the original ground state and the inferred spin state is minimized when we stop the process of quantum annealing before the amplitude of the transverse field reaches zero in contrast to the conventional prescription. This result means that finite quantum fluctuations compensate for the effects of noise, at least, to some extent. Analytical calculations using the infinite-range mean-field model support our conclusion qualitatively.
We consider a range of unconventional modifications to quantum annealing (QA), applied to an artificial trial problem with continuously tunable difficulty. In this problem, inspired by ``transverse field chaos'' in larger systems, classical and quantum methods are steered toward a false local minimum. To go from this local minimum to the global minimum, all NNN spins must flip, making this problem exponentially difficult to solve. We numerically study this problem by using a variety of new methods from the literature: inhomogeneous driving, adding transverse couplers, and other types of coherent oscillations in the transverse field terms (collectively known as RFQA, which stands for Random Field Quantum Annealing or Radio Frequency Quantum Annealing). We show that all of these methods improve the scaling of the time to solution (relative to the standard uniform sweep evolution) in at least some regimes. Comparison of these methods could help identify promising paths towards a demonstrable quantum speedup over classical algorithms in solving some realistic problems with near-term quantum annealing hardware.
We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. Quantum fluctuations cause transitions between states and thus play the same role as thermal fluctuations in the conventional approach. The idea is tested by the transverse Ising model, in which the transverse field is a function of time similar to the temperature in the conventional method. The goal is to find the ground state of the diagonal part of the Hamiltonian with high accuracy as quickly as possible. We have solved the time-dependent Schr\"odinger equation numerically for small size systems with various exchange interactions. Comparison with the results of the corresponding classical (thermal) method reveals that the quantum annealing leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule.
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.