Tight Success Probabilities for Quantum Period Finding and Phase Estimation

2025-06-25

AI Summary

This work addresses the success probability of quantum period-finding and phase estimation algorithms, establishing the first tight upper and lower bounds that converge to unity. Prior studies only provided constant lower bounds. Leveraging a novel integration of Fourier analysis, refined probabilistic estimation, and mathematical induction, we rigorously derive asymptotically optimal bounds on the success probability. Our analysis reveals that as the size of the quantum register increases, the success probability approaches one exponentially fast; moreover, the gap between the upper and lower bounds decays exponentially with the number of qubits. These results significantly enhance the theoretical characterization of the performance of these two foundational quantum algorithms, providing a tight analytical benchmark for reliability assessment and resource estimation in quantum computation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Evaluation and Analysis
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

Period finding and phase estimation are fundamental in quantum computing. Prior work has established lower bounds on their success probabilities. We improve these results by deriving tight upper and lower bounds on the success probability that converge to 1.

Problem

• Deriving tight bounds for quantum period finding success
• Improving phase estimation success probability limits
• Converging success probability bounds to 1

Innovation

• Derives tight success probability bounds
• Improves prior quantum period finding
• Converges bounds to optimal probability