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. We prove several results for this computational problem: - For the pure-state version, we show that the problem is BQP-hard for all nontrivial groups, and contained in QCMA $\cap$ QCSZK. We further obtain refined results for specific groups of interest: for abelian groups we show that the problem reduces to the state hidden subgroup problem over the generalized dihedral group; for the Clifford group, the problem is at least as hard as Graph Isomorphism under polynomial-time reductions; for the Pauli group it is BQP-complete. - For the mixed-state version, for nontrivial, finite and efficiently representable groups, the problem is QSZK-complete. - We also study a variant of this problem over an infinite group, in particular, the bosonic linear optical unitaries. We show that in the setting where the classical description of the quantum state is given in a suitable wave function representation known as the stellar representation, the problem is at least as hard as Graph Isomorphism, and is contained in NP $\cap$ SZK. Prior to our work, state isomorphism problems had only been studied for the symmetric group [LG17]. As a consequence of our results, we resolve an open question posed in [HEC25] about the existence of a quantum algorithm for the abelian state hidden subgroup problem on mixed states. We show that this problem is QSZK-hard in the worst case, thereby ruling out an efficient quantum algorithm unless QSZK = BQP.

翻译：我们研究群作用下量子态同构问题的计算复杂性：给定两个制备纯态或混合态的量子电路，判定这两个态是否通过某个群作用相关联。这可以视为隐移位问题的量子态版本，正如态隐子群问题是普通隐子群问题的量子版本。我们针对该计算问题证明了若干结果： - 对于纯态版本，我们证明该问题对所有非平凡群是BQP-难的，且包含于QCMA ∩ QCSZK。进一步，我们对若干特定群得到精细化结果：对于阿贝尔群，我们证明该问题可归约到广义二面体群上的态隐子群问题；对于克利福德群，该问题在多项式时间归约下至少与图同构问题同等困难；对于泡利群，该问题是BQP-完全的。 - 对于混合态版本，对非平凡、有限且可高效表示的群，该问题是QSZK-完全的。 - 我们还研究了该问题在无限群上的变体，特别是玻色线性光学幺正群。我们证明，在量子态经典描述以适当波函数表示（称为恒星表示）给出的设定下，该问题至少与图同构问题同等困难，且包含于NP ∩ SZK。 在本工作之前，态同构问题仅针对对称群被研究过[LG17]。作为我们结果的一个推论，我们解决了[HEC25]中提出的一个关于混合态阿贝尔态隐子群问题是否存在量子算法的开放问题。我们证明该问题在最坏情况下是QSZK-难的，从而排除了高效量子算法存在的可能性（除非QSZK = BQP）。