Noise can speed Markov chain Monte Carlo estimation and quantum annealing.

Abstract

Carefully injected noise can speed the average convergence of Markov chain Monte Carlo (MCMC) estimates and simulated annealing optimization. This includes quantum annealing and the MCMC special case of the Metropolis-Hastings algorithm. MCMC seeks the solution to a computational problem as the equilibrium probability density of a reversible Markov chain. The algorithm must cycle through a long burn-in phase until it reaches equilibrium because the Markov samples are statistically correlated. The special injected noise reduces this burn-in period in MCMC. A related theorem shows that it reduces the cooling time in simulated annealing. Simulations showed that optimal noise gave a 76% speed-up in finding the global minimum in the Schwefel optimization benchmark. The noise-boosted simulations found the global minimum in 99.8% of trials compared with only 95.4% of trials in noiseless simulated annealing. Simulations also showed that the noise boost is robust to accelerated cooling schedules and that noise decreased convergence times by more than 32% under aggressive geometric cooling. Molecular dynamics simulations showed that optimal noise gave a 42% speed-up in finding the minimum potential energy configuration of an eight-argon-atom gas system with a Lennard-Jones 12-6 potential. The annealing speed-up also extends to quantum Monte Carlo implementations of quantum annealing. Noise improved ground-state energy estimates in a 1024-spin simulated quantum annealing simulation by 25.6%. The quantum noise flips spins along a Trotter ring. The noisy MCMC algorithm brings each Markov step closer on average to equilibrium if an inequality holds between two expectations. Gaussian or Cauchy jump probabilities reduce the noise-benefit inequality to a simple quadratic inequality. Simulations show that noise-boosted simulated annealing is more likely than noiseless annealing to sample high probability regions of the search space and to accept solutions that increase the search breadth.