Efficient Identification of Permutation Symmetries in Many-Body Hamiltonians via Graph Theory

TL;DR

This paper presents a graph-theoretic algorithm to efficiently identify the full permutation symmetry group of arbitrary Pauli Hamiltonians, facilitating symmetry exploitation in quantum many-body simulations.

Contribution

It introduces a novel method linking Hamiltonian symmetries to graph automorphisms, enabling polynomial-time symmetry detection for bounded locality Hamiltonians.

Findings

Algorithm correctly identifies known symmetries in tested models.

Polynomial-time complexity for bounded locality Hamiltonians.

Permutation equivalence decision reduces to graph isomorphism.

Abstract

The computational cost of simulating quantum many-body systems can often be reduced by taking advantage of physical symmetries. While methods exist for specific symmetry classes, a general algorithm to find the full permutation symmetry group of an arbitrary Pauli Hamiltonian is notably lacking. This paper introduces a new method that identifies this symmetry group by establishing an isomorphism between the Hamiltonian's permutation symmetry group and the automorphism group of a coloured bipartite graph constructed from the Hamiltonian. We formally prove this isomorphism and show that for physical Hamiltonians with bounded locality and interaction degree, the resulting graph has a bounded degree, reducing the computational problem of finding the automorphism group to polynomial time. The algorithm's validity is empirically confirmed on various physical models with known symmetries. We further show that the problem of deciding whether two Hamiltonians are permutation-equivalent is polynomial-time reducible to the graph isomorphism problem using our graph representation. This work provides a general, structurally exact tool for algorithmic symmetry finding, enabling the scalable application of these symmetries to Hamiltonian simulation problems.