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Springer proceedings in physics

Replica Optimization Method for Ground-State Search of Random Spin-Systems

Naoki Kawashima, Masuo Suzuki

Published Jan 1, 1992• 16 citations
Publisher
DOI
10.1007/978-3-642-84821-6_45

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The path-integral quantum Monte Carlo (PIQMC) method is widely used as a classical simulation algorithm for quantum annealing. Replicas represent different points in imaginary time. We propose and demonstrate a swarm-like ground-state searching algorithm based on the PIQMC method for automatically fluctuation-controlled simulated annealing. Replicas help one another to search for the ground state of an Ising system as if forming a swarm and working cooperatively. Their interactions are local, simple and fluctuate to a certain degree. Such fluctuations are necessary to escape local minima, but the fluctuations do not need to be explicitly controlled. The size of fluctuations is automatically adjusted as annealing proceeds. We solve max-cut problems for algorithm evaluation, each of which corresponds to a graph of 100 vertices. We also solve the same problems using a conventional method based on the Metropolis algorithm for comparison.

Nonlinear Theory and Its Applications IEICE•Jan 1, 2023•0 citations
A new method to find ground states is proposed for random spin systems. It is applicable to systems with any boundary conditions, any bond distribution and any magnetic field. The efficiency of this method is confirmed numerically in the case of the two-dimensional Ising spin glass with Gaussian bond distribution in a uniform field. The introduction of more than two replicas improves the efficiency of the method considerably. It is also found that the renormalization process is effective. The increase in computational time with respect to system size is moderate and well fitted by a power law up to L=32. Magnetizations are calculated for various magnetic field using the new method. The size dependence of the susceptibility is found to be chi (L) varies as L x with x=0.476(5). This is somewhat larger than predictions using domain-wall renormalization group arguments.
We consider the problem of predicting the spin states in a kinetic Ising model when spin trajectories are observed for only a finite fraction of sites. In a Bayesian setting, where the probabilistic model of the spin dynamics is assumed to be known, the optimal prediction can be computed from the conditional (posterior) distribution of unobserved spins given the observed ones. Using the replica method, we compute the error of the Bayes optimal predictor for parallel discrete time dynamics in a fully connected spin system with non-symmetric random couplings. The results, exact in the thermodynamic limit, agree very well with simulations of finite spin systems.
The path-integral quantum Monte Carlo (PIQMC) method is widely used as a classical simulation algorithm for quantum annealing. Replicas represent different points in imaginary time. We propose and demonstrate a swarm-like ground-state searching algorithm based on the PIQMC method for automatically fluctuation-controlled simulated annealing. Replicas help one another to search for the ground state of an Ising system as if forming a swarm and working cooperatively. Their interactions are local, simple and fluctuate to a certain degree. Such fluctuations are necessary to escape local minima, but the fluctuations do not need to be explicitly controlled. The size of fluctuations is automatically adjusted as annealing proceeds. We solve max-cut problems for algorithm evaluation, each of which corresponds to a graph of 100 vertices. We also solve the same problems using a conventional method based on the Metropolis algorithm for comparison.