Efficient Algorithms for Weakly-Interacting Quantum Spin Systems
January 30, 2026
Gabriel Waite and I have just uploaded to arXiv our paper "Efficient Algorithms for Weakly-Interacting Quantum Spin Systems". In this paper we establish efficient algorithms for weakly-interacting quantum spin systems at arbitrary temperature. In particular, we obtain a fully polynomial-time approximation scheme for the partition function and an efficient approximate sampling scheme for the thermal distribution over the classical spin space.
The cluster expansion provides a powerful tool for developing approximation algorithms for statistical mechanical systems. This method has been successfully applied to obtain efficient algorithms for various models, including the hardcore model, the Potts model, and quantum spin systems. For quantum spin systems, efficient algorithms have been established at high temperature and at low temperature for stable quantum perturbations of classical spin systems. Our work extends these results to weakly-interacting systems at arbitrary temperature.
We consider quantum spin systems modelled by a multihypergraph $G=(V,E)$ with Hamiltonians of the form $H_\Phi + \lambda H_\Psi$, where $H_\Phi$ is non-interacting, $H_\Psi$ is a local perturbation, and $\lambda\in\mathbb{C}$ is a parameter. The partition function at inverse temperature $\beta$ is defined by $Z_G(\beta,\lambda):=\operatorname{Tr}\left[e^{-\beta(H_\Phi + \lambda H_\Psi)}\right]$, and the thermal state by $\rho_G(\beta,\lambda):=\left(Z_G(\beta,\lambda)\right)^{-1}e^{-\beta(H_\Phi+\lambda H_\Psi)}$.
Our main result for the partition function is as follows.
As a corollary, we obtain an efficient approximate sampling scheme for the thermal distribution over the classical spin space.
These results complement recent work on efficient algorithms for approximating the ground state energy and correlation functions of weakly-interacting quantum spin systems, rapid mixing of thermal samplers for weakly-interacting quantum systems, and efficient algorithms for approximating partition functions of weakly-interacting fermionic systems.
