The partition function and free energy of a quantum many-body system determine its physical properties in thermal equilibrium. Here we study the computational complexity of approximating these quantities for $n$-qubit local Hamiltonians. First, we report a classical algorithm with $\mathrm{poly}(n)$ runtime which approximates the free energy of a given $2$-local Hamiltonian provided that it satisfies a certain denseness condition. Our algorithm combines the variational characterization of the free energy and convex relaxation methods. It contributes to a body of work on efficient approximation algorithms for dense instances of optimization problems which are hard in the general case, and can be viewed as simultaneously extending existing algorithms for (a) the ground energy of dense $2$-local Hamiltonians, and (b) the free energy of dense classical Ising models. Secondly, we establish polynomial-time equivalence between the problem of approximating the free energy of local Hamiltonians and three other natural quantum approximate counting problems, including the problem of approximating the number of witness states accepted by a QMA verifier. These results suggest that simulation of quantum many-body systems in thermal equilibrium may precisely capture the complexity of a broad family of computational problems that has yet to be defined or characterized in terms of known complexity classes. Finally, we summarize state-of-the-art classical and quantum algorithms for approximating the free energy and show how to improve their runtime and memory footprint.

翻译：摘要：量子多体系统的配分函数和自由能决定了其在热平衡状态下的物理性质。本文研究了近似计算$n$量子比特局域哈密顿量这些量的计算复杂度。首先，我们提出一种经典算法，其运行时间为$\mathrm{poly}(n)$，可在给定$2$-局域哈密顿量满足特定稠密性条件时近似其自由能。该算法结合了自由能的变分刻画与凸松弛方法，为稠密实例（通常情况下难以处理）的优化问题中高效近似算法的研究做出了贡献，可视为同时推广了（a）稠密$2$-局域哈密顿量的基态能量近似算法，以及（b）稠密经典Ising模型的自由能近似算法。其次，我们建立了局域哈密顿量自由能近似问题与其他三个自然量子近似计数问题（包括近似QMA验证器接受见证态数量的问题）之间的多项式时间等价性。这些结果表明，热平衡下量子多体系统的模拟可能精确刻画了一类广泛计算问题的复杂度，而此类问题尚未用已知复杂度类别定义或刻画。最后，我们总结了当前用于近似自由能的最优经典和量子算法，并展示了如何改进其运行时间和内存占用。