标题： 用最优样本预测量子动力学的对称性 显示英文标题

标题： Predicting symmetries of quantum dynamics with optimal samples

摘要： 在量子动力学中识别对称性，如身份对称性或时间反演不变性，是一个具有深远影响的关键挑战。 我们引入了一个统一的框架，结合群表示理论和子群假设检验，以最优效率预测这些对称性。 通过利用紧致群及其不可约表示的固有对称性，我们推导出最优II型错误（检测对称性的失败概率）的精确表征，为量子最大相对熵提供了操作性解释。 特别是，我们证明并行策略能达到与自适应或无限因果序协议相同的效果，解决了关于复杂控制序列必要性的争论。 应用于单例群、最大交换群和正交群，得到明确结果：对于预测未知量子比特幺正的 identity 属性、Z 对称性和 T 对称性，零 I 型错误和 II 型错误被$\delta$限制，我们建立了显式的最优样本复杂度，其规模为$\mathcal{O}(\delta^{-1/3})$用于 identity 测试，$\mathcal{O}(\delta^{-1/2})$用于 T/Z 对称性测试。 这些发现为量子信息处理中的高效幺正性质测试和对称性驱动的协议提供了理论见解和实践指导。 显示英文摘要

摘要： Identifying symmetries in quantum dynamics, such as identity or time-reversal invariance, is a crucial challenge with profound implications for quantum technologies. We introduce a unified framework combining group representation theory and subgroup hypothesis testing to predict these symmetries with optimal efficiency. By exploiting the inherent symmetry of compact groups and their irreducible representations, we derive an exact characterization of the optimal type-II error (failure probability to detect a symmetry), offering an operational interpretation for the quantum max-relative entropy. In particular, we prove that parallel strategies achieve the same performance as adaptive or indefinite-causal-order protocols, resolving debates about the necessity of complex control sequences. Applications to the singleton group, maximal commutative group, and orthogonal group yield explicit results: for predicting the identity property, Z-symmetry, and T-symmetry of unknown qubit unitaries, with zero type-I error and type-II error bounded by $\delta$, we establish the explicit optimal sample complexity which scales as $\mathcal{O}(\delta^{-1/3})$ for identity testing and $\mathcal{O}(\delta^{-1/2})$ for T/Z-symmetry testing. These findings offer theoretical insights and practical guidelines for efficient unitary property testing and symmetry-driven protocols in quantum information processing.