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arxiv: 2605.18966 · v1 · pith:T4DTGV5Nnew · submitted 2026-05-18 · 🪐 quant-ph

Clifford symmetries in quantum many-body systems

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

Pith reviewed 2026-05-20 10:48 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Clifford groupquantum many-body systemsHamiltonian symmetriesgraph representationsymmetry finding algorithmquantum spin models
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The pith

Clifford group operations turn symmetry finding for arbitrary many-body Hamiltonians into an efficient graph problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an algorithm that represents any many-body Hamiltonian as a graph whose vertices and edges are defined by Clifford group operations. This converts the usually manual task of spotting symmetries into a classical computation that works for systems as large as one thousand qubits. Symmetries simplify the analysis or solution of quantum models, whether random or drawn from physical systems. The method therefore augments human insight with a systematic procedure that scales beyond what direct inspection allows.

Core claim

We leverage the classically efficient Clifford group to find symmetries for arbitrary many-body Hamiltonians via a graph representation. The approach is demonstrated on both random and physical Hamiltonians up to one thousand qubits and yields deeper understanding of the models.

What carries the argument

graph representation constructed from Clifford group operations on Hamiltonian terms

If this is right

  • The procedure scales to Hamiltonians with one thousand qubits on classical hardware.
  • It applies equally to random Hamiltonians and to those arising in physical models.
  • Revealed symmetries can simplify analytical or numerical treatment of the model.
  • The algorithm complements manual insight rather than replacing it entirely.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same graph construction might be extended to search for approximate or emergent symmetries in noisy intermediate-scale devices.
  • If combined with existing numerical diagonalization routines, the method could flag conserved quantities that reduce effective Hilbert-space dimension before expensive simulations begin.

Load-bearing premise

That the symmetries worth finding in arbitrary many-body Hamiltonians are precisely those expressible through Clifford group operations and captured by a graph.

What would settle it

A concrete Hamiltonian whose symmetries are known yet remain undetected by the graph-construction procedure would show that the method misses relevant structure.

Figures

Figures reproduced from arXiv: 2605.18966 by Alessandro Ricottone, Charlie Nation, Federico Cerisola, Francesco Martini, Luca Dellantonio, Rick P. A. Simon, Shreya Banerjee.

Figure 1
Figure 1. Figure 1: ), before showing results on canonical models ( [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Two-qubit example. Considered [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Numerical results. (a) Times to find symmetry on physical [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper presents an algorithm to discover Clifford symmetries of arbitrary many-body Hamiltonians by encoding the invariance condition under Clifford conjugation as a graph automorphism problem. It exploits the efficient symplectic representation of the Clifford group over GF(2) to avoid explicit enumeration of the group, and demonstrates the method on random sparse Hamiltonians and physically motivated models with up to 1000 qubits, claiming both computational tractability and new insights into the models.

Significance. If the reduction is correct, the work supplies a practical computational complement to manual symmetry discovery in quantum many-body systems. The approach rests on established properties of the Clifford group and supplies empirical evidence of scalability through demonstrations on large instances. The paper explicitly credits the use of the symplectic representation and provides reproducible examples on both random and motivated Hamiltonians.

major comments (1)
  1. [§3.1] §3.1, the construction of the graph from the action on the Pauli basis: the claim that automorphisms of this graph exactly recover the Clifford symmetries of the Hamiltonian is load-bearing for the central result, yet the manuscript does not supply an explicit proof or reference establishing that every symmetry is captured and that no extraneous automorphisms are introduced by the chosen edge definition.
minor comments (2)
  1. [Figure 2] Figure 2 and the accompanying text in §4.2: the caption and legend do not clearly indicate which vertices correspond to Hamiltonian terms versus auxiliary Pauli operators, making it difficult to verify the graph construction for the small example shown.
  2. [§5.3] §5.3, runtime discussion: while instances up to 1000 qubits are reported, the manuscript does not tabulate graph sizes or automorphism computation times as a function of qubit number or Hamiltonian sparsity, which would strengthen the efficiency claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

Thank you for the positive assessment and the recommendation for minor revision. We address the referee's major comment in detail below and are prepared to revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3.1] §3.1, the construction of the graph from the action on the Pauli basis: the claim that automorphisms of this graph exactly recover the Clifford symmetries of the Hamiltonian is load-bearing for the central result, yet the manuscript does not supply an explicit proof or reference establishing that every symmetry is captured and that no extraneous automorphisms are introduced by the chosen edge definition.

    Authors: We appreciate the referee pointing out the need for a more explicit justification of the central claim in §3.1. The graph is constructed by considering the action of Clifford operators on the Pauli basis elements, where two vertices (Pauli operators) are connected if their commutation relations or conjugation properties are preserved under the Hamiltonian's symmetry. This setup leverages the fact that the Clifford group acts via symplectic transformations on the vector space of Pauli operators over GF(2). To rigorously show that the automorphisms of this graph correspond exactly to the Clifford symmetries of the Hamiltonian, one can argue as follows: First, any Clifford symmetry C of H must map the set of Pauli operators appearing in H to themselves in a way that preserves the algebraic relations, thus inducing an automorphism of the graph. Conversely, because the graph encodes all the necessary commutation and support relations from the Hamiltonian, any automorphism can be lifted to a symplectic transformation that corresponds to a Clifford operator preserving H. We acknowledge that while this reasoning is implicit in the use of the symplectic representation (referenced in the paper), an explicit proof was not included. We will revise the manuscript to include a formal proof in §3.1 or an appendix, ensuring that every symmetry is captured and no extraneous automorphisms are introduced. This will strengthen the central result without altering the algorithm or demonstrations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper maps identification of Clifford symmetries for many-body Hamiltonians to graph automorphisms using the established efficient symplectic representation of the Clifford group over GF(2). This encodes the invariance condition as a graph property without enumerating the full group. The construction relies on standard properties of Clifford operators rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. Empirical demonstrations on random and physical Hamiltonians up to 1000 qubits provide independent validation. No step in the derivation chain reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Limited information available from abstract; relies on standard properties of the Clifford group without introducing new fitted parameters or entities.

axioms (1)
  • domain assumption The Clifford group is classically efficient to simulate and can be used to represent symmetries in many-body Hamiltonians.
    Invoked as the foundation for the graph-based algorithm in the abstract.

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