Monte Carlo studies of quantum and classical annealing on a double-well

TL;DR

This study compares classical and quantum Monte Carlo annealing methods on a double-well potential, highlighting the impact of move choices, tunneling challenges, and kinetic energy forms on optimization performance.

Contribution

It provides a comparative analysis of classical and quantum Monte Carlo annealing techniques, introducing a relativistic kinetic energy approach to improve quantum tunneling sampling.

Findings

Relativistic kinetic energy enhances quantum tunneling efficiency.

Move choice in classical annealing affects convergence and spectrum.

Path-Integral Monte Carlo faces sampling difficulties due to tunneling.

Abstract

We present results for a variety of Monte Carlo annealing approaches, both classical and quantum, benchmarked against one another for the textbook optimization exercise of a simple one-dimensional double-well. In classical (thermal) annealing, the dependence upon the move chosen in a Metropolis scheme is studied and correlated with the spectrum of the associated Markov transition matrix. In quantum annealing, the Path-Integral Monte Carlo approach is found to yield non-trivial sampling difficulties associated with the tunneling between the two wells. The choice of fictitious quantum kinetic energy is also addressed. We find that a ``relativistic'' kinetic energy form, leading to a higher probability of long real space jumps, can be considerably more effective than the standard one.