{"id":"93ed8b3e-8445-4e70-a993-6f6f7ea2b15d","arxiv_id":"2605.30428","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"An algorithm encodes Clifford invariants of qudit Hamiltonians as graph properties so graph automorphisms yield Clifford symmetries up to phase checks, tested on models and extended to open systems.","lead":"This paper maps finding Clifford symmetries in qudit Hamiltonians to a graph automorphism problem by encoding invariants as graph properties on Hamiltonian terms. A generalist might read it to learn how graph algorithms can automate symmetry detection in quantum models including open systems.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Graph encoding completeness for qudit Clifford invariants is the load-bearing assumption","rationale":"The reader's weakest_assumption directly identifies the same point; the full-text description of the encoding would be needed to move beyond this, but the load-bearing risk remains exactly where the reader located it.","tokens_in":1618,"tokens_out":277,"duration_ms":14992,"concrete_test":"From the algorithm section, extract the precise rules used to assign vertices and edges from a 3-qudit Pauli Hamiltonian; build the graph for the known symmetry group of the 3-qudit Heisenberg model, compute its automorphism group, apply the phase-correction procedure, and verify whether the resulting symmetries exactly match the independently enumerated Clifford symmetries of that model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the chosen vertex/edge labeling of Hamiltonian terms (encoding commutation relations, support overlaps, and Clifford invariants) is both complete and faithful: every graph automorphism must induce a valid Clifford symmetry (after phase checks), and every Clifford symmetry must arise this way. The abstract states the principle but supplies no explicit construction rules or proof that the encoding is bijective for general qudits; if the graph omits higher-order invariants or admits extra automorphisms, the GA output will either miss symmetries or produce invalid ones.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1728,"tokens_out":536,"duration_ms":14958,"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":[{"comment":"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.","section":"Abstract / algorithm description"},{"comment":"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.","section":"Tests / examples"}],"minor_comments":[{"comment":"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.","section":"Open systems paragraph"},{"comment":"The scaling discussion would benefit from a table comparing GA runtime versus brute-force enumeration for the tested models.","section":"Scaling analysis"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a direct follow-up to the authors' own recent preprint; the editor may wish to confirm that the novelty disclosure is adequate for the target journal."},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"[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."},{"response":"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_made":"yes","referee_comment":"[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."}],"tokens_in":1318,"tokens_out":491,"duration_ms":20779,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that the authors extend their earlier qubit work to give a concrete algorithm for locating Clifford symmetries in general qudit models, including open systems, by encoding invariants as graph properties and solving for automorphisms.\n\nWhat is new is the explicit procedure for labeling Hamiltonian terms as vertices for qudits, the handling of open-system cases, tests on several physical models, and the scaling discussion with number of qudits and Pauli strings plus optimization strategies for different regimes.\n\nThe paper does a solid job describing the algorithm steps and showing practical use on examples. The scaling and optimization sections add value for anyone who might implement this.\n\nThe soft spot is the completeness of the graph encoding itself. The central step assumes that every automorphism of the constructed graph corresponds to a valid Clifford symmetry (after phase checks) and that no symmetries are missed. This builds directly on their prior paper, so the base case is referenced, but the qudit extension needs clear rules showing the encoding is bijective for arbitrary dimension. If the full text supplies explicit labeling tables and verification on the test models, that would address it; otherwise the claim rests on the assumption holding without counterexamples.\n\nThis is aimed at quantum information people who need computational tools to find symmetries in qudit codes or simulations. A reader who wants a working method rather than new theory will get something usable from it.\n\nIt deserves a serious referee because it supplies a new algorithm with tests and scaling data. I would send it to peer review.","headline":"This paper gives an algorithm to find Clifford symmetries in qudit Hamiltonians by mapping to graph automorphism and extends it to open systems.","tokens_in":2192,"tokens_out":377,"would_cite":false,"duration_ms":19032,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Clifford symmetries of qudit Hamiltonians are recovered exactly by graph automorphisms on a graph whose vertices encode the Hamiltonian terms and their invariants.","keywords":["Clifford symmetries","graph automorphisms","qudit Hamiltonians","open quantum systems","Lindblad master equation","Pauli strings"],"falsifier":"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.","tokens_in":2532,"feed_emoji":"","tokens_out":474,"duration_ms":16628,"temperature":0.7,"pith_summary":"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.","feed_headline":"Graph automorphisms recover Clifford symmetries for qudits","feed_subtitle":"Encoding Hamiltonian terms as vertices turns symmetry search into a standard graph computation that also covers open systems.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Graph automorphisms map Hamiltonian terms to Clifford symmetries","Encoding qudit Hamiltonians as graphs yields Clifford symmetries","Graph automorphisms solve for Clifford symmetries in open qudit models","Clifford symmetries emerge from graph automorphisms in qudit systems"],"cache_read_input_tokens":64,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Graph automorphisms map Hamiltonian terms to Clifford symmetries","Encoding qudit Hamiltonians as graphs yields Clifford symmetries","Graph automorphisms solve for Clifford symmetries in open qudit models","Clifford symmetries emerge from graph automorphisms in qudit systems"]},"model":"grok-4.3","cost_usd":0.002786,"raw_usage":{"total_tokens":1503,"prompt_tokens":562,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":27862000,"prompt_tokens_details":{"text_tokens":562,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":883,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":562,"tokens_out":58,"duration_ms":6969,"temperature":1.0,"reasoning_tokens":883,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T06:36:23.223173+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}