Hypothesis testing of symmetry in quantum dynamics

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 six and five queries, respectively, from which we infer that the type-II error exhibits a decay rate of $O({m}^{\ensuremath{-}2})$ 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 $O({m}^{\ensuremath{-}1})$.