Pith Number

pith:PJ3GE75U

pith:2026:PJ3GE75U2BR42BWCDMSKKB3I4W

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Quantum state isomorphism problems for groups

Alexandru Gheorghiu, Arsalan Motamedi, Dale Jacobs, Saeed Mehraban

The quantum state isomorphism problem for mixed states under nontrivial finite group actions is QSZK-complete.

arxiv:2605.12615 v1 · 2026-05-12 · quant-ph · cs.CC

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

51 extracted · 51 resolved · 12 Pith anchors

[1] Physical Review A , volume= 2014

[2] Is Quantum Mechanics An Island In Theoryspace? · arXiv:quant-ph/0401062

[3] arXiv preprint arXiv:2102.05227 , year=

[4] Proceedings of the 57th Annual ACM Symposium on Theory of Computing , pages=

[5] arXiv preprint arXiv:2505.15770 , year=

Receipt and verification

First computed	2026-05-18T03:10:00.528172Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

7a76627fb4d063cd06c21b24a50768e5a10812083df674ab600ae3e8c780d1e6

Aliases

arxiv: 2605.12615 · arxiv_version: 2605.12615v1 · doi: 10.48550/arxiv.2605.12615 · pith_short_12: PJ3GE75U2BR4 · pith_short_16: PJ3GE75U2BR42BWC · pith_short_8: PJ3GE75U

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/PJ3GE75U2BR42BWCDMSKKB3I4W \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 7a76627fb4d063cd06c21b24a50768e5a10812083df674ab600ae3e8c780d1e6

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "3863f4a07e6d057d6cbc4851740c6b7892f0cee2a8e163873ed0509b8d240000",
    "cross_cats_sorted": [
      "cs.CC"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "quant-ph",
    "submitted_at": "2026-05-12T18:05:27Z",
    "title_canon_sha256": "6c1c045ba8389d262518c310a1e687f7a711fe008856477b39a0d254f12995fa"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12615",
    "kind": "arxiv",
    "version": 1
  }
}