发表机构
Instituto de Física Teórica UAM/CSIC（UAM/CSIC 理论物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种马尔可夫链算法，在连通团簇空间中直接采样团簇展开，避免了符号问题，实现了对量子多体系统配分函数的高效多项式时间计算。
AI 中文摘要
符号问题仍然是阻碍蒙特卡洛方法推广到一般量子多体系统的主要障碍。一个强有力的替代方案依赖于团簇展开，它为自由能及相关量提供了收敛的表示，其应用范围从张量网络收缩到量子动力学和电路。在其收敛范围内，这些展开通常能产生确定性的多项式时间算法。然而，此类方法需要对连通团簇进行穷举枚举，导致计算开销为$(N\varepsilon^{-1})^{\mathcal{O}(\log\Delta)}$，其中$N$是系统尺寸，$\varepsilon$是所需精度，$\Delta$是相互作用图的度数。这在长程系统中尤为严重，此时$\Delta\equiv N$，导致准多项式运行时间。在此，我们引入一种马尔可夫链算法，直接在连通团簇空间中采样团簇展开，用蒙特卡洛采样取代确定性枚举。由于采样发生在抽象的团簇空间而非物理构型空间，在收敛范围内不存在传统的指数符号问题。对于一般聚合物模型，我们建立了运行时间为$\tilde{\mathcal{O}}(N^4\varepsilon^{-3})$的FPRAS。对于温度$\beta\leq\beta_{\mathrm{sampling}}\equiv\mathcal{O}(1)$下的量子配分函数，该时间改进为$\tilde{\mathcal{O}}(N^{3+1/D}\varepsilon^{-3})$，其中$D$是晶格维度。该框架直接适用于相互作用$\propto1/r^\alpha$的长程量子系统，包括$\alpha>2D$的$2$-局域模型。
英文摘要
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 $(N\varepsilon^{-1})^{\mathcal{O}(\logΔ)}$, where $N$ is the system size, $\varepsilon$ the desired precision, and $Δ$ the interaction-graph degree. This becomes particularly severe in long-range systems, where $Δ\equiv N$, 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 $\tilde{\mathcal{O}}(N^4\varepsilon^{-3})$. For quantum partition functions at temperatures $β\leqβ_{\mathrm{sampling}}\equiv\mathcal{O}(1)$, this improves to $\tilde{\mathcal{O}}(N^{3+1/D}\varepsilon^{-3})$, where $D$ is the lattice dimension. The framework directly applies to long-range quantum systems with interactions $\propto1/r^α$, including $2$-local models with $α>2D$.
