arXiv subjects

Ziyuan Dong

Publications and source records attributed to Ziyuan Dong.

2 recordsLinked to original sources

arXiv2025

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

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.

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arXiv2025

Probabilistic Bounds on the Number of Elements to Generate Finite Nilpotent Groups and Their Applications

This work establishes a new probabilistic bound on the number of elements to generate finite nilpotent groups. Let $φ_k(G)$ denote the probability that $k$ random elements generate a finite nilpotent group $G$. For any $0 < ε< 1$, we prove that $φ_k(G) \ge 1 - ε$ if $k \ge \operatorname{rank}(G) + \lceil \log_2(2/ε) \rceil$ (a bound based on the group rank) or if $k \ge \operatorname{len}(G) + \lceil \log_2(1/ε) \rceil$ (a bound based on the group chain length). Moreover, these bounds are shown to be nearly tight. Both bounds sharpen the previously known requirement of $k \ge \lceil \log_2 |G| + \log_2(1/ε) \rceil + 2$. Our results provide a foundational tool for analyzing probabilistic algorithms, enabling a better estimation of the iteration count for the finite Abelian hidden subgroup problem (AHSP) standard quantum algorithm and a reduction in the circuit repetitions required by Regev's factoring algorithm.

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