Fast Algorithms for Stoquastic Spin Systems
August 21, 2026
I have just uploaded to arXiv my paper "Fast Algorithms for Stoquastic Spin Systems". In this paper we establish a general framework for developing fast sampling and counting algorithms for stoquastic spin systems at high temperature. We apply our framework to obtain fast algorithms for approximating the partition function and sampling from the thermal distribution of general stoquastic spin systems, ferromagnetic Heisenberg models, and antiferromagnetic Heisenberg models on bipartite graphs.
The polymer model formalism provides a general framework for developing efficient algorithms for these problems. The principal algorithmic technique within this framework is the cluster expansion, which gives polynomial-time algorithms, though the degree of the polynomial is often relatively high. For classical systems, fast algorithms have been developed via Markov chain methods for polymer models. These fast classical algorithms are based on sampling directly from the polymer model rather than explicitly computing a truncated cluster expansion. In particular, the polymer dynamics of Chen et al. provides a rapidly mixing Markov chain for sampling polymer configurations, while the graphlet sampling algorithm of Blanca et al. provides an efficient method for sampling individual polymers via a subcritical percolation process. Our framework is based on these methods.
A stoquastic spin system $\mathcal{S}$ is modelled by a graph $G$. At each vertex $v$ of $G$, there is a $d$-dimensional Hilbert space $\mathcal{H}_v$ with $d<\infty$. The Hilbert space on the graph is given by $\mathcal{H}_G:=\bigotimes_{v \in V(G)}\mathcal{H}_v$. An interaction $\Phi$ assigns a self-adjoint operator $\Phi(e)$ on $\bigotimes_{v \in e}\mathcal{H}_v$ to each edge $e$ of $G$. The Hamiltonian of $\mathcal{S}$ is defined by $H_\mathcal{S}:=\sum_{e \in E(G)}\!\Phi(e)$. We consider stoquastic Hamiltonians in the sense that all matrix elements of $\Phi(e)$ are non-positive in the spin basis for all $e \in E(G)$. Let $\mathcal{S}=(\Phi,d,\beta)$ denote such a system at inverse temperature $\beta$.
Our main result for general stoquastic systems is as follows.
For the Heisenberg models, we obtain an improved bound on the inverse temperature by using the cycle representation of Tóth and the loop representation of Aizenman and Nachtergaele. Let $\mathcal{S}=(\beta,h)$ denote a Heisenberg model with inverse temperature $\beta\geq0$ and field strength $h\in\mathbb{R}$. Our main result for the ferromagnetic Heisenberg model is as follows.
Our main result for the antiferromagnetic Heisenberg model on bipartite graphs is as follows.
