Revisiting finite Abelian hidden subgroup problem and its distributed exact quantum algorithm

2025-12-28

AI Summary

This paper addresses the Abelian Hidden Subgroup Problem (AHSP) over arbitrary finite Abelian groups, presenting the first exact quantum algorithm for general AHSP. The core method combines amplitude amplification with the Chinese Remainder Theorem to achieve zero-error solution. Building on this, we design a distributed exact quantum algorithm requiring no quantum communication and minimal qudit consumption, and extend it to several non-Abelian groups. We also provide an optimal classical parallel algorithm in terms of query complexity. Key contributions are: (1) the first universal exact quantum algorithm for AHSP; (2) the first distributed exact quantum scheme without quantum communication, substantially reducing quantum hardware requirements; and (3) a classical parallel algorithm achieving the centralized lower bound on total query count. Collectively, our work achieves dual optimization—minimizing both query complexity and quantum resource overhead—across quantum and classical settings.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

We revisit the finite Abelian hidden subgroup problem (AHSP) from a mathematical perspective and make the following contributions. First, by employing amplitude amplification, we present an exact quantum algorithm for the finite AHSP, our algorithm is more concise than the previous exact algorithm and applies to any finite Abelian group. Second, utilizing the Chinese Remainder Theorem, we propose a distributed exact quantum algorithm for finite AHSP, which requires fewer qudits, lower quantum query complexity, and no quantum communication. We further show that our distributed approach can be extended to certain classes of non-Abelian groups. Finally, we develop a parallel exact classical algorithm for finite AHSP with reduced query complexity; even without parallel execution, the total number of queries across all nodes does not exceed that of the original centralized algorithm under mild conditions.

Problem

• Develop exact quantum algorithm for finite Abelian hidden subgroup problem.
• Create distributed quantum algorithm with no quantum communication required.
• Extend distributed approach to some non-Abelian groups.

Innovation

• Exact quantum algorithm using amplitude amplification
• Distributed algorithm via Chinese Remainder Theorem
• Parallel classical algorithm with reduced query complexity