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.
Proceedings of the Royal Society of London
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Quantum state isomorphism problems for groups
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.