{"paper":{"title":"Quantum state isomorphism problems for groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The quantum state isomorphism problem for mixed states under nontrivial finite group actions is QSZK-complete.","cross_cats":["cs.CC"],"primary_cat":"quant-ph","authors_text":"Alexandru Gheorghiu, Arsalan Motamedi, Dale Jacobs, Saeed Mehraban","submitted_at":"2026-05-12T18:05:27Z","abstract_excerpt":"We study the computational complexity of quantum state isomorphism problems under group actions: given two quantum circuits that prepare pure or mixed states, decide whether the two states are related by a group action. This can be seen as a quantum state version of the Hidden Shift Problem, in much the same way that the State Hidden Subgroup Problem is a quantum version of the ordinary Hidden Subgroup Problem.\n  We prove several results for this computational problem:\n  - For the pure-state version, we show that the problem is BQP-hard for all nontrivial groups, and contained in QCMA $\\cap$ Q"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For the mixed-state version, for nontrivial, finite and efficiently representable groups, the problem is QSZK-complete. We show that the abelian state hidden subgroup problem on mixed states is QSZK-hard in the worst case, thereby ruling out an efficient quantum algorithm unless QSZK = BQP.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The input states are given by quantum circuits; groups are nontrivial, finite, and efficiently representable (or use stellar representation for the bosonic case); standard quantum circuit model and group action definitions hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Quantum state isomorphism under group actions is BQP-hard for pure states across nontrivial groups and QSZK-complete for mixed states with finite groups; Pauli group version is BQP-complete and Clifford is GI-hard, ruling out efficient quantum algorithms for abelian mixed-state HS unless QSZK=BQP.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The quantum state isomorphism problem for mixed states under nontrivial finite group actions is QSZK-complete.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d9f68d3e38ce8f94f89b89e4e5e642e533e89e0c9c57707c93594daeb1606b00"},"source":{"id":"2605.12615","kind":"arxiv","version":1},"verdict":{"id":"f2a1b6c5-14ac-4ae7-a769-7682346f47c0","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:27:43.329225Z","strongest_claim":"For the mixed-state version, for nontrivial, finite and efficiently representable groups, the problem is QSZK-complete. We show that the abelian state hidden subgroup problem on mixed states is QSZK-hard in the worst case, thereby ruling out an efficient quantum algorithm unless QSZK = BQP.","one_line_summary":"Quantum state isomorphism under group actions is BQP-hard for pure states across nontrivial groups and QSZK-complete for mixed states with finite groups; Pauli group version is BQP-complete and Clifford is GI-hard, ruling out efficient quantum algorithms for abelian mixed-state HS unless QSZK=BQP.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The input states are given by quantum circuits; groups are nontrivial, finite, and efficiently representable (or use stellar representation for the bosonic case); standard quantum circuit model and group action definitions hold.","pith_extraction_headline":"The quantum state isomorphism problem for mixed states under nontrivial finite group actions is QSZK-complete."},"references":{"count":51,"sample":[{"doi":"","year":2014,"title":"Physical Review A , volume=","work_id":"25957a12-7503-47ac-a3d9-1e6b53e6edcf","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Is Quantum Mechanics An Island In Theoryspace?","work_id":"d391bae6-f58e-43f1-ab15-c9a14cb4f415","ref_index":2,"cited_arxiv_id":"quant-ph/0401062","is_internal_anchor":true},{"doi":"","year":null,"title":"arXiv preprint arXiv:2102.05227 , year=","work_id":"ba313dd1-5c58-4dc7-974e-4821a0340a58","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Proceedings of the 57th Annual ACM Symposium on Theory of Computing , pages=","work_id":"0260aaac-2182-4432-ba99-f958a640eb4d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"arXiv preprint arXiv:2505.15770 , year=","work_id":"1f543f3c-459c-44a0-9a51-343b52b14849","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":51,"snapshot_sha256":"6c103b61cbf9e4414170d04a1d3dc8a8b7cdddcfc199750b6fe73aea864caa12","internal_anchors":12},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}