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

2025-11-28

AI Summary

Efficiently identifying permutation symmetries of Pauli Hamiltonians in quantum many-body systems remains challenging due to the combinatorial complexity of symmetry group enumeration. Method: We propose a general algorithm based on colored bipartite graph modeling. Our approach rigorously establishes a group isomorphism between the permutation symmetry group of a Pauli Hamiltonian and the automorphism group of its associated colored bipartite graph, thereby reducing symmetry identification to a polynomial-time graph automorphism problem. Leveraging bounded locality of interactions, we construct low-degree graphs and integrate efficient automorphism detection techniques. Contribution/Results: The algorithm exactly and efficiently computes the full permutation symmetry group for arbitrary Pauli Hamiltonians. We validate its correctness and universality across standard physical models—including Heisenberg, XY, and toric code Hamiltonians—demonstrating substantial reduction in computational overhead for symmetry-driven quantum simulations.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Identifying full permutation symmetry groups in arbitrary Pauli Hamiltonians
• Reducing computational cost of quantum simulations through symmetry exploitation
• Establishing isomorphism between Hamiltonian symmetries and graph automorphisms

Innovation

• Mapping Hamiltonian symmetries to graph automorphisms
• Proving polynomial-time complexity for bounded-degree systems
• Reducing Hamiltonian equivalence to graph isomorphism problem