Tight Bound for Quantum Unitary Time-Reversal

2025-07-08

AI Summary

This work addresses the query complexity of unitary time reversal in the quantum black-box model: given oracle access to an unknown $d$-dimensional unitary $U$, what is the minimal number of queries required to approximate its inverse $U^{-1}$? Method: Employing the quantum query model and diamond-norm error analysis, the study incorporates average-case error characterization and exploits the phase independence of controlled-$U$ operations. Contribution/Results: We establish the first tight lower bound $Omega((1-varepsilon)d^2)$ for coherent protocols with unbounded ancillary systems, valid under diamond-norm error $varepsilon$; this bound extends to the average-case setting. Additionally, we derive a tight $Omega(d^2)$ lower bound for implementing controlled-$U$. These results resolve the fundamental resource cost of unitary time reversal in quantum learning and metrology, revealing an inherent quadratic query overhead for high-dimensional unitary inversion.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

Time-reversal of unitary evolution is fundamental in quantum information processing. Many scenarios, particularly those in quantum learning and metrology, assume free access to the time-reverse of an unknown unitary. In this paper, we settle the query complexity of the unitary time-reversal task: approximately implementing $U^{-1}$ given only black-box access to an unknown $d$-dimensional unitary $U$. We provide a tight query lower bound $Ω((1-ε)d^2)$ for the unitary time-reversal to within diamond norm error $ε$. Notably, our lower bound applies to general coherent protocols with unbounded ancillas, and holds even when $ε$ is an average-case distance error. Moreover, our result implies a query lower bound $Ω(d^2)$ for approximately implementing control-$U$ up to an irrelevant phase, which is also tight with respect to the dimension.

Problem

• Determining query complexity for unitary time-reversal
• Establishing tight lower bound for implementing U⁻¹
• Extending lower bound to control-U implementation

Innovation

• Tight query lower bound for unitary time-reversal
• Applies to coherent protocols with ancillas
• Lower bound holds for average-case error