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

Saumya Shah, Patrick Rebentrost

TL;DR

<3-5 sentence high-level summary>The paper tackles the challenge of identifying the full permutation symmetry group of arbitrary Pauli Hamiltonians by introducing a graph-theoretic isomorphism: the Hamiltonian's permutation symmetry group G_H is isomorphic to the automorphism group of a carefully constructed coloured bipartite graph aut(G). It proves the isomorphism and shows that, for physically motivated bounded-locality Hamiltonians, the resulting graph has bounded degree, enabling polynomial-time computation of the automorphism group with existing GA/GI tools. The approach is validated on several Ising and Heisenberg models, demonstrating that the computed automorphism generators reproduce known symmetry groups, and the method is shown to faithfully reduce permutation equivalence to graph isomorphism. This provides a scalable, exact tool for discovering and exploiting permutation symmetries in Hamiltonian simulation and permutation-equivariant circuit design.

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.

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

TL;DR

<3-5 sentence high-level summary>The paper tackles the challenge of identifying the full permutation symmetry group of arbitrary Pauli Hamiltonians by introducing a graph-theoretic isomorphism: the Hamiltonian's permutation symmetry group G_H is isomorphic to the automorphism group of a carefully constructed coloured bipartite graph aut(G). It proves the isomorphism and shows that, for physically motivated bounded-locality Hamiltonians, the resulting graph has bounded degree, enabling polynomial-time computation of the automorphism group with existing GA/GI tools. The approach is validated on several Ising and Heisenberg models, demonstrating that the computed automorphism generators reproduce known symmetry groups, and the method is shown to faithfully reduce permutation equivalence to graph isomorphism. This provides a scalable, exact tool for discovering and exploiting permutation symmetries in Hamiltonian simulation and permutation-equivariant circuit design.

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.

Paper Structure

This paper contains 44 sections, 3 theorems, 27 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Symmetries in Hamiltonians
  4. Exploiting Symmetries in Hamiltonians.
  5. Algorithmic Identification of Symmetries in Hamiltonians.
  6. The Graph Automorphism Problem
  7. Unaddressed Research Gaps
  8. Mathematical Preliminaries
  9. Symmetries
  10. Hamiltonian.
  11. Symmetries.
  12. Permutation Symmetry.
  13. Graph Automorphisms and Isomorphisms
  14. Transverse Field Ising Model
  15. 1D TFIM
  16. ...and 29 more sections

Key Result

Proposition 1

The permutation symmetry group of the Hamiltonian $\mathcal{H}$ is isomorphic to the automorphism group of the graph $\mathcal{G}$ constructed from $\mathcal{H}$ as Sec. ssec:constr.

Figures (2)

  • Figure 1: A square lattice corresponding to a 2D TFIM
  • Figure 2: Coloured Bipartite Graph for Eq. \ref{['eq:exampleforgraph']}. The graph is constructed by creating nodes for the qubits and the terms in a bipartite manner, and adding edges between a term and a qubit if the term acts non-trivially on the qubit. Then, a colouring is assigned to the nodes and edges as described in Sec. \ref{['ssec:constr']}.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof