Probabilistic Bounds on the Number of Elements to Generate Finite Nilpotent Groups and Their Applications
Ziyuan Dong, Xiang Fan, Tengxun Zhong, Daowen Qiu
TL;DR
This work addresses the problem of how many random elements are needed to generate a finite nilpotent group with high probability. It introduces two sharp bounds based on intrinsic group structure: $k \ge \operatorname{rank}(G) + \lceil \log_2(2/\\epsilon) \rceil$ or $k \ge \operatorname{len}(G) + \lceil \log_2(1/\\epsilon) \rceil$, guaranteeing $\\varphi_k(G) \ge 1-\\epsilon$; the authors prove these bounds via a Sylow decomposition and reduce to bounds for elementary abelian factors, with near-tightness demonstrated by concrete counterexamples. The results sharpen prior universal bounds and yield direct improvements in quantum algorithms: (i) tighter iteration counts for the finite Abelian hidden subgroup problem, and (ii) a reduction in circuit repetitions in Regev's factoring algorithm, by leveraging the relationship $H^{\\perp} \\cong G/H$ and the additivity of chain length. Overall, the paper provides a foundational probabilistic tool for analyzing generation in finite nilpotent groups and demonstrates meaningful practical gains in quantum and probabilistic computation contexts.
Abstract
This work establishes a new probabilistic bound on the number of elements to generate finite nilpotent groups. Let $\varphi_k(G)$ denote the probability that $k$ random elements generate a finite nilpotent group $G$. For any $0 < ε< 1$, we prove that $\varphi_k(G) \ge 1 - ε$ 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.