Predicting symmetries of quantum dynamics with optimal samples

TL;DR

This paper develops an optimal framework for predicting symmetries in quantum dynamics using group theory and hypothesis testing, providing explicit sample complexities for various symmetry tests.

Contribution

It introduces a unified, optimal approach combining group representation theory and hypothesis testing to efficiently identify quantum symmetries, resolving debates on protocol complexity.

Findings

Parallel strategies match adaptive protocols in performance.

Explicit sample complexities for identity and symmetry testing.

Optimal sample complexity scales as $oldsymbol{ ext{O}( ext{delta}^{-1/3})}$ and $oldsymbol{ ext{O}( ext{delta}^{-1/2})}$.

Abstract

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.