Clock Factorized Quantum Monte Carlo Method for Long-range Interacting Systems

TL;DR

This work tackles the high computational cost of simulating long-range interacting quantum systems, where standard local updates incur $\mathcal{O}(N)$ work per move. It introduces clock factorized QMC, a framework that factorizes the Metropolis filter into independent interaction-term factors and uses recursive clock sampling with bound hazards to achieve $\mathcal{O}(1)$ typical update costs. The authors develop three clock-factorized algorithms—Clock Factorized Metropolis, Clock Factorized Worm, and Clock Factorized Worm with Long-range Hopping—for LRTFIM, EBHM, and LRXXZ, respectively, demonstrating substantial efficiency gains in 2D and 3D systems and robust applicability to both diagonal and off-diagonal long-range terms. The method leverages a tree-structured sampling of first-rejection events, bound hazards, and dynamic thinning to enable fast, flexible updates across various update schemes, suggesting broad impact for long-range interacting physics. Overall, clock factorized QMC provides a practical and general tool for exploring complex quantum systems with long-range couplings, with potential applications ranging from Coulombic and dipolar interactions to quantum simulations.

Abstract

Simulating long-range interacting systems is a challenging task due to its computational complexity that the computational effort for each local update is of order $\cal{O}$$(N)$, where $N$ is the size of system. Recently, a technique, called hereby the clock factorized quantum Monte Carlo method, was developed on the basis of the so-called factorized Metropolis filter [Phys. Rev. E 99 010105 (2019)]. In this work, we first explain step by step how the clock factorized quantum Monte Carlo method is implemented to reduce the computational overhead from $\cal{O}$$(N)$ to $\cal{O}$(1). In particular, the core ingredients, including the concepts of bound probabilities and bound rejection events, the tree-like data structure, and the fast algorithms for sampling an extensive set of discrete and small probabilities, are elaborated. Next, we show how the clock factorized quantum Monte Carlo method can be flexibly implemented in various update strategies, like the Metropolis and worm-type algorithms, and can be generalized to simulate quantum systems. Finally, we demonstrate the high efficiency of the clock factorized quantum Monte Carlo algorithms in the examples of the quantum Ising model and the Bose-Hubbard model with long-range interactions and/or long-range hopping amplitudes. We expect that the clock factorized quantum Monte Carlo algorithms would find broad applications in statistical and condensed-matter physics.

Figures (5)

• Figure 1: (a)In the clock sampling process, one determines the fate of a proposed update by sampling clocks from the probability distribution $C(X), X \in [0,N]$. Each clock represents a possible outcome of the factorized Metropolis filter. The first $N-1$ clocks is the first rejection events and the clock hand points the first rejecting factor. The last clock is the acceptance events where all factors permit the update. (b) Schematic illustration of the recursive sampling process of the first-bound-rejection events on a tree structure.
• Figure 2: An example of the alias method for a discrete distribution of $4$ elements. For a target distribution $p(k)$, a possible alias table is shown.
• Figure 3: A schematic diagram of one level of the clock sampling. The blue box represents the hazard rate $h_j$ of factors, and the gray box represents the corresponding bound hazard rate $\hat{h}_j$. Starting from the $j$-th factor, the first step (I) generates a jump to the $j'$-th factor using a geometric random number with parameter $\rho$. The second step (II) is to accept $j'$ as a bound-rejection event with relative probability $\hat{h}_{j'}/\rho$. If $j'$ is rejected, then one goes back to (I). Otherwise, if $j'$ is indeed a bound-rejection event, then one goes to the third step (III) to check if factor $j'$ truly rejects the update with $h_{j'}/\hat{h}_{j'}$
• Figure 4: Average complexity of Monte Carlo update of 2D models. (a) long-range transverse field Ising model using Metropolis update, (b) extended Bose-Hubbard model using worm update, (c) long-range XXZ model using worm update with long-range hopping.
• Figure 5: Average complexity of Monte Carlo update of 3D models. (a) long-range transverse field Ising model using Metropolis update, (b) extended Bose-Hubbard model using worm update, (c) long-range XXZ model using worm update with long-range hopping.