We develop randomized matrix-free algorithms for estimating partial traces, a generalization of the trace arising in quantum physics and chemistry. Our algorithm improves on the typicality-based approach used in [T. Chen and Y-C. Cheng, \emph{Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems}, J. Chem. Phys. 157, 064106 (2022)] by deflating important subspaces (e.g. corresponding to the low-energy eigenstates) explicitly. This results in a significant variance reduction, leading to several order-of-magnitude speedups over the previous state of the art. We then apply our algorithm to study the thermodynamics of several Heisenberg spin systems, particularly the entanglement spectrum and ergotropy.

翻译：我们开发了用于估计偏迹的随机化无矩阵算法，偏迹是量子物理与化学中迹的一种推广。我们的算法通过显式收缩重要子空间（例如对应低能本征态的子空间），改进了[T. Chen and Y-C. Cheng, \emph{Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems}, J. Chem. Phys. 157, 064106 (2022)]中使用的基于典型性的方法。这实现了显著的方差缩减，从而相比先前最优方法获得了数个数量级的加速。随后，我们将算法应用于研究若干海森堡自旋系统的热力学性质，特别是纠缠谱与功容。