Predicting symmetries of quantum dynamics with optimal samples

2025-02-03

AI Summary

This work addresses the efficient detection of quantum state symmetries—specifically identity, time-reversal (T), and Z-symmetry—under time-reversal invariance, minimizing the sample complexity required to achieve a prescribed confidence level while precisely characterizing failure probability and quantum max-relative entropy. Method: We develop a unified framework integrating group representation theory with subgroup hypothesis testing. Contribution/Results: We derive the first exact optimal bound on Type-II error for symmetry testing; prove the equivalence—in sample complexity—of parallel, adaptive, and indefinite-causal-order strategies; and establish tight lower bounds on sample complexity for all considered symmetries. Under zero Type-I error constraint, we achieve sample complexities of $O(delta^{-1/3})$ for identity testing and $O(delta^{-1/2})$ for T- and Z-symmetry testing on qubit unitary operations—providing theoretically optimal benchmarks for quantum device characterization and symmetry-driven protocols.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

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.

Problem

• Quantum Symmetry Detection
• Time Reversal Invariance
• Minimum Sample Size

Innovation

• Symmetry Detection
• Time-Reversal Invariance
• Quantum Information Processing