Microcanonical simulated annealing: Massively parallel Monte Carlo simulations with sporadic random-number generation

Numerical simulations of models and theories that describe complex systems such as spin glasses are becoming increasingly important. Beyond fundamental research, these computational methods also find practical applications in fields like combinatorial optimization. However, Monte Carlo simulations, an important subcategory of these methods, are plagued by a major drawback: they are extremely greedy for (pseudo) random numbers. The total fraction of computer time dedicated to random-number generation increases as the hardware grows more sophisticated, and can get prohibitive for special-purpose computing platforms. We propose here a general-purpose microcanonical simulated annealing (MicSA) formalism that dramatically reduces such a burden. The algorithm is fully adapted to a massively parallel computation, as we show in the particularly demanding benchmark of the three-dimensional Ising spin glass. We carry out very stringent numerical tests of the new algorithm by comparing our results, obtained on GPUs, with high-precision standard (i.e., random-number-greedy) simulations performed on the Janus II custom-built supercomputer. In those cases where thermal equilibrium is reachable (i.e., in the paramagnetic phase), both simulations reach compatible values. More significantly, barring short-time corrections, a simple time rescaling suffices to map the MicSA off-equilibrium dynamics onto the results obtained with standard simulations.