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arXiv:2605.30428

Graph automorphisms to obtain Clifford symmetries in open and closed qudit models

Charlie Nation, Rick P. A. Simon, Shreya Banerjee, Francesco Martini, Alessandro Ricottone, Federico Cerisola, Luca Dellantonio

quant-phSubmitted May 28, 2026• Announced Jun 1, 2026• 0 citations
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Abstract

In the recent article [arXiv:2605.18966], we demonstrated that finding Clifford symmetries can be mapped to a Graph Automorphism (GA) problem. Here, we provide an algorithm to obtain such symmetries on general qudit systems, that works on the principle of encoding Clifford invariants of a Hamiltonian onto properties of a graph. Labelling Hamiltonian terms as vertices, a permutation of such vertices that respects the Clifford invariants (a GA) is both a valid Clifford, and a symmetry up to phase correction checks. We test this on multiple physical models and discuss the scaling with respect to the number of qudits and Pauli strings, as well as various strategies for optimisation in different regimes. We further show that the graph automorphism representation of Clifford symmetries can be expanded to open quantum systems.

Comments
10 + 6 pages. Companion article to arXiv:2605.18966. Comments welcome

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In the recent article [arXiv:2605.18966], we demonstrated that finding Clifford symmetries can be mapped to a Graph Automorphism (GA) problem. Here, we provide an algorithm to obtain such symmetries on general qudit systems, that works on the principle of encoding Clifford invariants of a Hamiltonian onto properties of a graph. Labelling Hamiltonian terms as vertices, a permutation of such vertices that respects the Clifford invariants (a GA) is both a valid Clifford, and a symmetry up to phase correction checks. We test this on multiple physical models and discuss the scaling with respect to the number of qudits and Pauli strings, as well as various strategies for optimisation in different regimes. We further show that the graph automorphism representation of Clifford symmetries can be expanded to open quantum systems.
Gauge theories are important descriptions for many physical phenomena and systems in quantum computation. Automorphism of gauge group naturally gives global symmetries of gauge theories. In this work we study such symmetries in gauge theories induced by automorphisms of the gauge group, when the gauge theories have nontrivial topological actions in different spacetime dimensions. We discover the automorphism symmetry can be extended, become a higher group symmetry, and/or become a non-invertible symmetry. We illustrate the discussion with various models in field theory and on the lattice. In particular, we use automorphism symmetry to construct new transversal non-Clifford logical gates in topological quantum codes. In particular, we show that 2+1d ZN\mathbb{Z}_NZN​ qudit Clifford stabilizer models can implement non-Clifford transversal logical gate in the 4th level ZN\mathbb{Z}_NZN​ qudit Clifford hierarchy for N≥3N\geq 3N≥3, extending the generalized Bravyi-König bound proposed in the companion paper [arXiv:2511.02900] for qubits.
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.
NOTE: PAPER WITHDRAWN (See Comments) The Clifford and Local Clifford groups for d>2d > 2d>2 dimensional systems have been topics of recent interest due to their applications in graph states, quantum codes, and possible applications in fast quantum algorithms. This paper studies these groups more abstractly, by first characterizing the Pauli Automorphism and Local Automorphism groups, and then using these results to determine characteristics of the Clifford and Local Clifford groups. Not only does such an approach reveal new information about the Clifford and Local Clifford groups, but it also shows how many previously derived results arise naturally as simple corollaries. Lastly, we give a systematic method of building an arbitrary Local Clifford operator from a small number of previously known gates, as well as a method to physically implement such an operator.
Obtaining the symmetries of a model is a critical step towards developing an understanding and ultimately analytically or numerically solving the model. However, finding symmetries is generally extremely complicated, often being the result of insightful thinking. In this work, we complement human ingenuity with an algorithm. We leverage the classically efficient Clifford group to find symmetries for arbitrary many-body Hamiltonians via a graph representation. We demonstrate our method on random and physical Hamiltonians, with instances of up to one thousand qubits and demonstrate how our approach can provide deeper understanding of the model.
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