Accelerating Multicanonical Sampling with Irreversibility

TL;DR

This work tackles the slow, diffusive convergence of flat-histogram Monte Carlo methods by introducing irreversible lifting into multicanonical sampling. By partitioning moves into directed, energy-change-based chains and allowing controlled inter-chain jumps via a lifting parameter, the method creates biased energy-space flows while preserving global balance. The approach yields significant speedups in ground-state searches and energy-range exploration for both the 2D Ising model and Edwards--Anderson spin glasses, with reduced round-trip variability and maintained accuracy, albeit with modest per-move overhead. These results suggest practical impact for large-scale ground-state problems and hint at effective integration with parallel multicanonical schemes in statistical physics and related optimization tasks.

Abstract

Flat-histogram Monte Carlo simulations are well-established, robust methods to perform random walks in a physical observable or parameter space, making them suitable for finding ground states or studying phase transitions in complex systems in statistical physics. However, their efficiency can be limited by the time to attain the desired flat distribution, which is generally unknown prior to the simulations. In particular, they might suffer from slowing down towards the end of a simulation due to the diffusive nature of random walks. In this work we apply irreversibility to the multicanonical Monte Carlo method via the lifting approach to alleviate this behavior. We achieve a 2-4 times speedup in ground-state search for a two-dimensional (2D) Ising model, and up to an order of magnitude of speedup for finding the ground-state energy in an Edwards-Anderson spin glass, compared to traditional multicanonical sampling. The round-trip times between ground states show a narrower distribution and are significantly shorter compared to the reversible counterpart, suggesting that a lower convergence time with a smaller time variance is feasible.

Figures (9)

• Figure 1: a) Transitions between states are equally likely in both directions (detailed balance), a walk through those states is always "reversible". b) Transitions between states are only possible in one directions, yet the probability "influx" equals the "outflux" in each state (global balance).
• Figure 2: The lifting scheme for a 2D Ising model consisting of separate chains corresponding to changes in energy $|\Delta E|=0,4,$ and $8$. Each $|\Delta E| \neq 0$ chain has a direction, $\sigma = +1$ or $\sigma= -1$, where a MC move results in an energy change; whereas on the $|\Delta E|=0$ chain a MC move does not result in an energy change. Trial moves are only proposed on the current chain and in the current direction of the sampling. Switching chains and direction happens in separate steps.
• Figure 3: Logarithm of the density of states for $L=64$ and $L=128$ as estimated by lifted multicanonical sampling at $\theta=1$ (solid lines) compared with the known, exact solution (symbols, shown at intervals for clarity). The inset shows the statistical error, $\varepsilon$, from multiple runs and the absolute deviation, $\Delta$, of their average from the exact solution ($\Delta=\lvert\overline{\ln g(E)}-\ln g_{\mathrm{exact}}(E)\rvert$) for $L=128$.
• Figure 4: Kullback--Leibler divergence between the true logarithmic density of states of the $L=128$ 2D Ising model and the logarithm of the instantaneous, estimated density of states as a function of Monte-Carlo time. Different curves compare the convergence of "lifted" multicanonical simulations at different lifting parameters $\theta$ with the conventional multicanonical sampling without lifting. Error bars were obtained from independent runs for each setting.
• Figure 5: Top: Time evolution of the lowest energy found for the $L=64$ Ising model at different lifting parameters, showing how the explored energy range widens over time. The upper boundary of the explored energy range (at positive energies) is symmetric to the lower boundary and not shown. Three individual runs are shown for each value of $\theta$ to show the consistency of the method, the curves end when the ground state is found for the first time. Bottom: The average Monte-Carlo (MC) time needed to find the ground state for the first time depending on the lifting parameter. The horizontal dashed line denotes the MC time needed for the original multicanonical algorithm to find the ground state for the first time.