Abstract The partition function is an essential quantity in statistical mechanics, and its accurate computation is a key component of any statistical analysis of quantum systems and phenomena. However, for interacting many-body quantum systems, its calculation generally involves summing over an exponential number of terms and can thus quickly grow to be intractable. Accurately and efficiently estimating the partition function of its corresponding system Hamiltonian then becomes the key in solving quantum many-body problems. In this paper we develop a hybrid quantum–classical algorithm to estimate the partition function, utilising a novel quantum Clifford sampling technique. Note that previous works on the estimation of partition functions require O ( 1 / ϵ Δ ) -depth quantum circuits (Srinivasan et al 2021 IEEE Int. Conf. on Quantum Computing and Engineering (QCE) pp 112–22; Montanaro 2015 Proc. R. Soc. A 471 20150301), where Δ is the minimum spectral gap of stochastic matrices and ϵ is the multiplicative error. Our algorithm requires only a shallow O ( 1 ) -depth quantum circuit, repeated O ( n / ϵ 2 ) times, to provide a comparable ϵ approximation. Shallow-depth quantum circuits are considered vitally important for currently available noisy intermediate-scale quantum devices.

中文翻译：

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用量子 Clifford 采样估计 Gibbs 配分函数

摘要 配分函数是统计力学中的一个基本量，其精确计算是任何量子系统和现象统计分析的关键组成部分。然而，对于相互作用的多体量子系统，其计算通常涉及对指数数量的项求和，因此可能很快变得难以处理。准确有效地估计其对应系统哈密顿量的配分函数成为解决量子多体问题的关键。在本文中，我们开发了一种混合量子经典算法来估计配分函数，利用一种新的量子克利福德采样技术。其中 Δ 是随机矩阵的最小谱间隙，ε 是乘法误差。我们的算法只需要一个浅 O ( 1 ) 深度的量子电路，重复 O ( n / ϵ 2 ) 次，以提供可比较的 ϵ 近似值。浅深度量子电路被认为对于当前可用的嘈杂中等规模量子器件至关重要。