Long-Range Bosonic Systems at Thermal Equilibrium: Computational Complexity and Clustering of Correlations

TL;DR

This work proves that long-range bosonic systems, specifically the Bose-Hubbard model with couplings decaying as r^{-α} with α > D, admit efficient classical algorithms for approximating thermal partition functions at high temperature, achieving a quasi-polynomial runtime e^{O(log^{2}(N/ε))} while introducing a controllable poly(N)^{-1} error due to Hilbert-space truncation. It also establishes a rigorous power-law clustering of correlations at high temperature, |C_{β}(O_X,O_Y)| ≤ C^{|X|+|Y|} Φ_{β}(O_X,O_Y)/(1+d_{X,Y})^{α}, and a thermal area law, using a novel interaction-picture cluster expansion that correctly handles unbounded bosonic local Hilbert spaces. A low-boson-density inequality justifies truncating local Hilbert spaces to a finite cutoff q ∝ log N, enabling practical computation and providing a rigorous error bound. The paper also contrasts long-range and finite-range bosonic systems, showing that finite-range models admit an almost-polynomial-time algorithm with slightly different error structure due to truncation, and discusses broader implications for quantum statistical mechanics and computational complexity in bosonic many-body systems.

Abstract

Long-range systems, characterized by couplings that decay as a power law $r^{-α}$, are of fundamental importance and attract widespread interest across diverse physical phenomena. Among these, bosonic systems are particularly significant due to their theoretical importance and experimental relevance. In this Letter, we propose a classical algorithm with a quasipolynomial runtime to efficiently approximate the partition function of long-range bosonic systems at high temperatures. For finite-range systems, the complexity improves to almost polynomial time. We also present a rigorous proof for the power-law decay of correlation functions. This property, known as clustering of correlation, is well-established for finite-range spin models. However, in sharp contrast, has remained largely unexplored for long-range bosonic systems. The results presented here address two long-standing gaps concerning the computational complexity and correlation clustering in such systems. The methodology we introduce provides new tools for future studies of challenging problems in statistical and quantum many-body physics concerning bosonic system and computational complexity.

Figures (7)

• Figure 1: Illustration of the clustering theorem for long-range bosonic systems. Summarized as Theorem \ref{['theorem_boson_lr_clustering']}, the magnitude of the correlation function $C_{\beta}(O_{X},O_{Y})$ exhibits a power-law decay with distance $d_{X,Y}$ at high temperatures.
• Figure 2: Computational complexities for approximating the partition function of quantum lattice models at high temperatures. This table compares state-of-art results for finite- and long-range spin systems (see Refs. mann2024algorithmic and sanchez2025high) with this work's primary results for bosons, which are formally stated in Theorems \ref{['theorem_boson_lr_complexity']} and \ref{['theorem_boson_fr_complexity']}, respectively. In this context, $N$ denotes the system size, and $\epsilon > 0$ is the desired precision.
• Figure 3: This figure provides a schematic overview of the main results, methodological framework, and structure of this supplementary material, which concerns the study of high-temperature thermal states in long-range bosonic systems. Our analysis approaches the problem from the perspectives of both complexity theory and statistical mechanics. The system size here is denoted by $N \coloneq |V|$.
• Figure 4: An example where the multiset $G = \{Z_1, Z_2, \ldots, Z_8\}$ connects $X$ and $Y$, but $G\oplus X\oplus Y$ is not connected.
• Figure 5: Illustration of the concept of a "simply connected" multiset.

Theorems & Definitions (92)

• Theorem 1
• Remark 1
• Theorem 2
• Remark 2
• Theorem 3
• Remark 3
• Corollary 1
• Theorem 4
• Definition 1: Connect/Connected Multiset
• Lemma 1