发表机构
Max-Planck-Institut für Quantenoptik; Munich Center for Quantum Science and Technology (MCQST); Dahlem Center for Complex Quantum Systems, Freie Universität Berlin（马克斯·普朗克量子光学研究所; 慕尼黑量子科学与技术中心; 柏林自由大学达勒姆复杂量子系统中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了阿贝尔态隐藏子群问题在样本与查询模型下的最优复杂度，证明查询模型因相干访问制备电路而获得ε的二次加速，并给出稳定子群学习等应用。
AI 中文摘要
在寻找具有超多项式加速的进一步量子算法的探索中，一个反复出现的主题是：问题的复杂度在很大程度上由输入访问模型决定。在此，我们针对态隐藏子群问题（StateHSP）研究这一现象，该问题是隐藏子群问题的量子推广，其目标是识别未知量子态的对称性。对于有限阿贝尔群，现有的傅里叶采样算法使用 $O(\log(|G|)/\epsilon)$ 份态副本，但这一规模是否最优一直悬而未决。我们在先前研究的样本模型和一种新的查询模型中解决了阿贝尔StateHSP的复杂度问题，后者是一种更强且操作上自然的推广，提供对态制备酉算子及其逆的访问。在查询模型中，我们给出一个时间高效的量子算法，使用 $O(\log(|G/H|)/\sqrt{\epsilon})$ 次正向和逆向查询，并证明匹配的 $\Omega(\log(|G/H|)/\sqrt{\epsilon})$ 下界，该下界即使在更强的共轭查询和受控查询设置中也成立。相比之下，我们表明在样本模型中，$\Theta(\log(|G/H|)/\epsilon)$ 份副本既是充分的，也是信息论上必要的，即使允许任意集体测量也是如此。因此，在 $\epsilon$ 上的二次改进确实源于对制备电路的相干访问。作为应用，我们获得了学习稳定子群、定位非纠缠态和识别隐藏平移对称性的更快算法。
英文摘要
In the quest to identify further quantum algorithms exhibiting superpolynomial speed-ups, a recurring theme is that the complexity of a problem is largely shaped by the input access model. Here, we study this phenomenon for the state hidden subgroup problem (StateHSP), a quantum generalization of the hidden subgroup problem in which the goal is to identify the symmetries of an unknown quantum state. For finite abelian groups, existing Fourier-sampling algorithms use $O(\log(|G|)/ε)$ copies of the state, but whether this scaling is optimal has remained open. We settle the complexity of the abelian StateHSP in both the previously studied sample model and a new query model, which is a stronger and operationally natural generalization that provides access to the state-preparation unitary and its inverse. In the query model, we give a time-efficient quantum algorithm using $O(\log(|G/H|)/\sqrtε)$ forward and inverse queries, and prove a matching $Ω(\log(|G/H|)/\sqrtε)$ lower bound which holds even in the stronger conjugate-query and controlled-query settings. By contrast, we show that in the sample model, $Θ(\log(|G/H|)/ε)$ copies are both sufficient and information-theoretically necessary, even if one allows for arbitrary collective measurements. Thus, the quadratic improvement in $ε$ genuinely arises from coherent access to the preparation circuit. As applications, we obtain faster algorithms for learning stabilizer groups, locating unentanglement, and identifying hidden translation symmetries.
