Optimal Convergence Rate of Lie-Trotter Approximation for Quantum Thermal Averages

• URL: http://arxiv.org/abs/2309.05188v5
• Date: Mon, 20 Oct 2025 14:21:07 GMT
• Title: Optimal Convergence Rate of Lie-Trotter Approximation for Quantum Thermal Averages
• Authors: Xuda Ye, Zhennan Zhou,
• Abstract summary: Lie--Trotter product formula is a foundational approximation for the quantum partition function.<n>This paper provides a quantitative error analysis for this approximation across two key systems.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The Lie--Trotter product formula is a foundational approximation for the quantum partition function, yet obtaining rigorous error bounds for the unbounded Hamiltonians common in physics remains a significant challenge. This paper provides a quantitative error analysis for this approximation across two key systems. For a particle in a smooth, periodic potential, we establish an optimal convergence rate of $\mathcal O(1/N^2)$ for both the partition function and thermal averages, where $N$ is the number of imaginary time steps. We then extend this analysis to the more challenging case of a confining potential on $\mathbb R$, proving a nearly optimal rate of $\mathcal O((\log N+1)^{\frac32}/N^2)$. The derived error bounds provide a firm mathematical foundation for the high-order accuracy of path integral simulations in quantum statistical mechanics.

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