Symmetry plays a crucial role in quantum physics, dictating the behavior and dynamics of physical systems. In this paper, we develop a hypothesis-testing framework for quantum dynamics symmetry using a limited number of queries to the unknown unitary operation and establish the quantum max-relative entropy lower bound for the type-II error. We construct optimal ancilla-free protocols that achieve optimal type-II error probability for testing time-reversal symmetry (T-symmetry) and diagonal symmetry (Z-symmetry) with limited queries. Contrasting with the advantages of indefinite causal order strategies in various quantum information processing tasks, we show that parallel, adaptive, and indefinite causal order strategies have equal power for our tasks. We establish optimal protocols for T-symmetry testing and Z-symmetry testing for 6 and 5 queries, respectively, from which we infer that the type-II error exhibits a decay rate of $\mathcal{O}(m^{-2})$ with respect to the number of queries $m$. This represents a significant improvement over the basic repetition protocols without using global entanglement, where the error decays at a slower rate of $\mathcal{O}(m^{-1})$.

翻译：对称性在量子物理学中扮演着关键角色，它决定了物理系统的行为与动力学。本文针对量子动力学对称性，利用对未知酉操作的有限次查询，建立了一个假设检验框架，并确立了第二类错误的量子最大相对熵下界。我们构建了无需辅助系统的最优协议，该协议在有限次查询下，能够以最优的第二类错误概率检验时间反演对称性（T-对称性）与对角对称性（Z-对称性）。与不确定因果序策略在各种量子信息处理任务中的优势形成对比，我们证明了对于我们的任务，并行、自适应以及不确定因果序策略具有同等效力。我们分别针对6次和5次查询，建立了T-对称性与Z-对称性检验的最优协议，并由此推断第二类错误相对于查询次数 $m$ 呈现出 $\mathcal{O}(m^{-2})$ 的衰减速率。相较于不使用全局纠缠的基本重复协议中误差以较慢的 $\mathcal{O}(m^{-1})$ 速率衰减，这代表了一个显著的改进。