发表机构
Imperial College London; Institute of Physics, Slovak Academy of Sciences（帝国理工学院; 斯洛伐克科学院物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出局部松弛层级与消息传递算法，为量子基态能量提供高效下界估计，并证明其收敛性，在实验中展现出超越标准凸优化求解器的潜力。
AI 中文摘要
凸松弛层级为量子多体系统的基态能量提供下界，这些下界可在经典计算机上以多项式时间计算，且层级固定。然而，由于传统求解器的计算成本以及效率保证的稀缺性，将这些方法扩展到大型系统并实现精确近似仍然具有挑战性。在本工作中，我们开发了局部松弛层级以及高效、高度可并行化的消息传递算法，用于估计松弛后的基态能量。我们证明了层级的第一级——基于成对约化密度矩阵的局部一致性——对于树上的交换哈密顿量是精确的。我们进一步建立了另一个基于一致区间的层级，对于链上可分离哈密顿量的弱扰动，该层级以区间大小的指数速度收敛到基态能量，从而为这些系统提供了一种高效的经典算法。然后，我们引入了两种消息传递算法的变体，对于有界度图上的局部层级的任何固定级别，它们分别以$\mathcal{O}(n/\epsilon^2)$和$\mathcal{O}(n/\epsilon)$的时间运行，其中$\epsilon$是每个位点松弛基态能量的精度。这假设最优消息具有$\mathcal{O}(1)$范数——我们在实验中的实际设置中观察到这一条件。这些算法基于次梯度方法和Nesterov型加速梯度下降方法，应用于熵平滑目标。最后，我们在不同的量子哈密顿量、晶格几何形状和松弛级别上对消息传递算法进行了基准测试，验证了理论预测及其在解决此问题上超越标准凸优化求解器的潜力。我们在以下网址发布了由此产生的库：此http URL。
英文摘要
Convex relaxation hierarchies provide lower bounds to the ground state energy of quantum many-body systems that can be computed in polynomial time on a classical computer, at any fixed hierarchy level. However, scaling these methods to large systems and accurate approximations remains challenging due to the computational cost of traditional solvers and the scarcity of efficiency guarantees. In this work, we develop local relaxation hierarchies and efficient, highly parallelisable message passing algorithms for estimating the relaxed ground state energies. We show that the first level of the hierarchy---based on local consistency of pairwise reduced density matrices---is exact for commuting Hamiltonians on trees. We further establish that another hierarchy, based on consistent intervals, converges exponentially fast in the interval size to the ground state energy for weak perturbations of separable Hamiltonians on a chain, thereby providing an efficient classical algorithm for these systems. Then, we introduce two variants of message passing algorithms that run in $\mathcal{O}(n/ε^2)$ and $\mathcal{O}(n/ε)$ time for any fixed level of the local hierarchy on bounded-degree graphs, where $ε$ is the precision for the relaxed ground state energy per site. This assumes that the optimal messages have $\mathcal{O}(1)$ norm---a condition we observe in practical settings in our experiments. These algorithms are based on the subgradient method and the Nesterov-type accelerated gradient descent method applied to an entropy-smoothed objective. Finally, we benchmark the message passing algorithms across different quantum Hamiltonians, lattice geometries, and relaxation levels, validating the theoretical predictions and their potential to surpass standard convex optimisation solvers for this problem. We release the resulting library at github.com/rick1924/gse-message-passing.
Comments39 pages, 4 figures
