Efficient Algorithms for Weakly-Interacting Quantum Spin Systems

TL;DR

This work develops efficient algorithms for weakly-interacting quantum spin systems at arbitrary temperature by leveraging the cluster expansion within an abstract polymer-model framework on bounded-degree bounded-rank multihypergraphs. It proves a fully polynomial-time approximation scheme for the partition function $Z_G(\beta,\lambda)$ and an efficient approximate sampler for the thermal distribution over classical spins, under a bound on the perturbation parameter $|\lambda|$. The methods rely on representing the partition function via an abstract polymer model, ensuring polymer weights decay suitably, and reducing sampling to counting through marginal probabilities. The results extend to fermionic settings and offer a concrete path toward scalable quantum-spin simulations in the weakly-interacting regime, with implications for ground-state energy estimation and thermal properties.

Abstract

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 a classical spin space. Our approach is based on the cluster expansion method and a standard reduction from approximate sampling to approximate counting.

Theorems & Definitions (14)

• Theorem 1
• Remark
• Lemma 2: restate=[name=restatement]PartitionFunctionAbstractPolymerModel
• Lemma 3: restate=[name=restatement]PartitionFunctionWeightBound
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['theorem:ApproximationAlgorithmPartitionFunction']}
• Theorem 5
• Remark
• Lemma 6: restate=[name=restatement]ApproximateCountingApproximateSampling