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 \mathcal{O}(N) 𝒪 ( N ) , where N N is the size of the system. In this work, we introduce the clock factorized quantum Monte Carlo method, an efficient technique for simulating long-range interacting quantum systems. The method is developed by generalizing the clock Monte Carlo method for classical systems [Phys. Rev. E 99 010105 (2019)] to the path-integral representation of long-range interacting quantum systems, with some specific treatments for quantum cases and a few significant technical improvements in general. We first explain how the clock factorized quantum Monte Carlo method is implemented to reduce the computational overhead from \mathcal{O}(N) 𝒪 ( N ) to \mathcal{O}(1) 𝒪 ( 1 ) . In particular, the core ingredients, including the concepts of bound probabilities and bound rejection events, the recursive sampling procedure, 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. Finally, we demonstrate the high efficiency of the clock factorized quantum Monte Carlo algorithms using examples of three typical long-range interacting quantum systems, including the transverse field Ising model with long-range z z - z z interaction, the extended Bose-Hubbard model with long-range density-density interactions, and the XXZ Heisenberg model with long-range spin interactions. We expect that the clock factorized quantum Monte Carlo method would find broad applications in statistical and condensed-matter physics.
The sign problem remains a major obstacle to extending the success of Monte Carlo methods to generic quantum many-body systems. A powerful alternative relies on cluster expansions, which provide convergent representations of free energies and related quantities, with applications ranging from tensor network contractions to quantum dynamics and circuits. Within their convergence regime, these expansions often yield deterministic polynomial-time algorithms. However, such methods require exhaustive enumeration of connected clusters, leading to a computational overhead of
(Nε−1)O(logΔ), where
N is the system size,
ε the desired precision, and
Δ the interaction-graph degree. This becomes particularly severe in long-range systems, where
Δ≡N, resulting in quasi-polynomial runtimes. Here, we introduce a Markov chain algorithm that samples cluster expansions directly in the space of connected clusters, replacing deterministic enumeration by Monte Carlo sampling. Because sampling occurs in an abstract cluster space rather than a physical configuration space, the conventional exponential sign problem is absent within the convergence regime. For general polymer models, we establish an FPRAS with runtime
O~(N4ε−3). For quantum partition functions at temperatures
β≤βsampling≡O(1), this improves to
O~(N3+1/Dε−3), where
D is the lattice dimension. The framework directly applies to long-range quantum systems with interactions
∝1/rα, including
2-local models with
α>2D.