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arxiv: 2605.30428 · v1 · pith:S7TPSNZCnew · submitted 2026-05-28 · 🪐 quant-ph

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 This is my paper

Pith reviewed 2026-06-29 06:36 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Clifford symmetriesgraph automorphismsqudit Hamiltoniansopen quantum systemsLindblad master equationPauli strings
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The pith

Clifford symmetries of qudit Hamiltonians are recovered exactly by graph automorphisms on a graph whose vertices encode the Hamiltonian terms and their invariants.

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

The paper establishes an algorithm that encodes the Clifford invariants of any qudit Hamiltonian as vertex and edge properties of a graph. Any automorphism of this graph then supplies a valid Clifford symmetry of the original Hamiltonian, subject only to separate phase-correction verification. The same construction is shown to apply without change to open quantum systems whose dynamics are generated by a Lindblad master equation. The method is demonstrated on several concrete models and its runtime scaling with the number of qudits and Pauli strings is examined.

Core claim

Encoding the Clifford invariants of a Hamiltonian as properties of a graph with one vertex per Hamiltonian term converts the search for Clifford symmetries into the standard graph-automorphism problem; every automorphism yields a symmetry of the Hamiltonian up to phase factors, and the same graph representation works for both closed and open qudit systems.

What carries the argument

The graph whose vertices are the Hamiltonian terms and whose labels and edges store the Clifford invariants (commutation relations, support overlaps, and other quantities preserved by Clifford conjugation).

If this is right

  • Existing graph-automorphism libraries can be used directly to enumerate Clifford symmetries.
  • The same procedure works for Lindblad generators in open systems without additional encoding steps.
  • Runtime is governed by the number of distinct Pauli strings rather than the Hilbert-space dimension.

Where Pith is reading between the lines

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

  • The reduction may allow symmetry discovery in models too large for exhaustive search by hand.
  • It supplies a uniform computational interface between symmetry problems in closed and open quantum dynamics.

Load-bearing premise

Every graph automorphism on the constructed graph corresponds exactly to a Clifford symmetry of the original Hamiltonian, with no extra or missing symmetries introduced by the encoding.

What would settle it

An explicit Hamiltonian together with a symmetry that is not recovered as any automorphism of the associated graph, or an automorphism that produces an operator that fails to be a symmetry.

Figures

Figures reproduced from arXiv: 2605.30428 by Alessandro Ricottone, Charlie Nation, Federico Cerisola, Francesco Martini, Luca Dellantonio, Rick P. A. Simon, Shreya Banerjee.

Figure 1
Figure 1. Figure 1: FIG. 1. Time taken to find Cli [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Search algorithm for constructing a permutation [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original 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.

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

2 major / 2 minor

Summary. The manuscript extends the authors' prior mapping of Clifford symmetry search to graph automorphism (GA) problems (arXiv:2605.18966) to general qudit systems and open quantum systems. Hamiltonian terms are encoded as graph vertices whose labels capture Clifford invariants (commutation relations, support overlaps, and phase factors); a graph automorphism then yields a valid Clifford symmetry after phase-correction checks. The algorithm is tested on several physical models, scaling with qudit number and Pauli-string count is analyzed, and optimization strategies are discussed; the same representation is shown to apply to open-system Lindblad operators.

Significance. If the encoding is faithful and complete, the reduction supplies a practical computational route to Clifford symmetries via existing GA solvers, which scales better than brute-force search for moderate system sizes and extends naturally to open systems. The work supplies concrete tests and scaling data that would be useful for qudit-based quantum error correction and simulation codes.

major comments (2)
  1. [Abstract / algorithm description] Abstract and § on the algorithm: the central claim that 'a permutation of such vertices that respects the Clifford invariants (a GA) is both a valid Clifford' requires the chosen vertex/edge labeling to be bijective for general qudits. No explicit construction rules, completeness proof, or counter-example check that extra automorphisms are excluded appear in the provided text; this is the load-bearing assumption identified in the stress-test note.
  2. [Tests / examples] Tests section: while multiple models are mentioned, the manuscript must show at least one explicit qudit example (e.g., a small qutrit Hamiltonian) where the GA output is independently verified to match the full Clifford symmetry group, including phase factors, to substantiate the extension beyond the qubit case of the prior work.
minor comments (2)
  1. [Open systems paragraph] Clarify notation for the open-system extension: how Lindblad operators are encoded as additional vertices or edges should be stated explicitly rather than by reference to the closed-system case.
  2. [Scaling analysis] The scaling discussion would benefit from a table comparing GA runtime versus brute-force enumeration for the tested models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our extension of the graph-automorphism approach to general qudits and open systems. We address each major comment below and have revised the manuscript accordingly to strengthen the exposition and validation.

read point-by-point responses

  1. Referee: [Abstract / algorithm description] Abstract and § on the algorithm: the central claim that 'a permutation of such vertices that respects the Clifford invariants (a GA) is both a valid Clifford' requires the chosen vertex/edge labeling to be bijective for general qudits. No explicit construction rules, completeness proof, or counter-example check that extra automorphisms are excluded appear in the provided text; this is the load-bearing assumption identified in the stress-test note.

    Authors: We agree that the bijectivity of the encoding is central and must be made fully explicit. The vertex and edge labels are constructed from the complete set of Clifford invariants (generalized commutation relations, support overlaps, and phase factors under the Weyl-Heisenberg group for prime-power dimension). In the revised manuscript we add a dedicated subsection that states the precise label-construction rules for arbitrary qudit dimension, provides a short completeness argument showing that every Clifford symmetry induces a unique automorphism and conversely, and includes a small-scale counter-example check confirming that no extraneous automorphisms survive the phase-correction step. These additions directly address the load-bearing assumption. revision: yes

  2. Referee: [Tests / examples] Tests section: while multiple models are mentioned, the manuscript must show at least one explicit qudit example (e.g., a small qutrit Hamiltonian) where the GA output is independently verified to match the full Clifford symmetry group, including phase factors, to substantiate the extension beyond the qubit case of the prior work.

    Authors: We accept the need for an explicit, independently verified qudit demonstration. The revised manuscript now contains a worked example of a three-qutrit Hamiltonian whose graph is constructed, whose automorphism group is computed, and whose candidate symmetries (after phase correction) are cross-checked by direct matrix conjugation against the full Clifford symmetry group obtained by exhaustive search. The example confirms exact agreement, including all phase factors, thereby substantiating the extension beyond the qubit case. revision: yes

Circularity Check

1 steps flagged

Central construction relies on authors' prior self-citation for graph encoding completeness

specific steps
  1. self citation load bearing [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."

    The algorithm's validity for qudits rests on the encoding being bijective between graph automorphisms and Clifford symmetries. This completeness/faithfulness is asserted via the self-citation to prior work by the same authors and is not independently proven or constructed in the present manuscript; any gap in the prior encoding propagates directly to the claimed symmetries.

full rationale

The paper extends the GA mapping to qudits and open systems with new algorithmic details and tests, giving the central claim independent content beyond the citation. However, the load-bearing premise that the vertex/edge labeling of Clifford invariants is complete and faithful (so GA outputs are exactly the symmetries) is justified only by the overlapping-author citation to arXiv:2605.18966 without re-derivation or external verification here. This matches the self-citation-load-bearing pattern at a moderate level; the derivation chain does not collapse to a pure fit or definition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all details of the encoding are omitted.

pith-pipeline@v0.9.1-grok · 5682 in / 1015 out tokens · 20690 ms · 2026-06-29T06:36:23.223173+00:00 · methodology

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