Practical algorithm for simulating thermal pure quantum states
Wei-Bo He, Yun-Tong Yang, Hong-Gang Luo
TL;DR
The paper addresses the challenge of benchmarking quantum many-body systems at finite temperature, where exact diagonalization is prohibitive and sign problems hinder QMC. It introduces a practical Thermal Pure Quantum (TPQ) state algorithm enhanced by a MatExp-based matrix exponential, implemented in the open-source Physica library, with symmetry-sector optimizations and grand-canonical extensions. The main contributions are a numerically stable TPQ approach that matches exact benchmarks more reliably than prior TPQ methods and a ~10^3× speedup on a 4×4 Hubbard model, enabling access to βt up to 32 and broader doping. This work provides a scalable, reproducible framework for finite-temperature benchmarking of strongly correlated quantum systems, accelerating validation of new algorithms and methods.
Abstract
The development of novel quantum many-body computational algorithms relies on robust benchmarking. However, generating such benchmarks is often hindered by the massive computational resources required for exact diagonalization or quantum Monte Carlo simulations, particularly at finite temperatures. In this work, we propose a new algorithm for obtaining thermal pure quantum states, which allows efficient computation of both mechanical and thermodynamic properties at finite temperatures. We implement this algorithm in our open-source C++ template library, Physica. Combining the improved algorithm with state-of-the-art software engineering, our implementation achieves high performance and numerical stability. As an example, we demonstrate that for the $4 \times 4$ Hubbard model, our method runs approximately $10^3$ times faster than $\mathcal{H}Φ$ 3.5.2. Moreover, the accessible temperature range is extended down to $β= 32$ across arbitrary doping levels. These advances significantly push forward the frontiers of benchmarking for quantum many-body systems.