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

TL;DR

This paper derives new probabilistic bounds on the number of elements needed to generate finite nilpotent groups, improving efficiency estimates for quantum algorithms like the Abelian hidden subgroup problem and Regev's factoring algorithm.

Contribution

It introduces tighter bounds based on group rank and chain length, nearly matching the theoretical minimum, enhancing analysis of probabilistic group generation.

Findings

Bounds are nearly tight and improve previous requirements

New bounds depend on group rank and chain length

Applications to quantum algorithms reduce iteration counts

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 $0 < \epsilon < 1$, we prove that $\varphi_k(G) \ge 1 - \epsilon$ if $k \ge \operatorname{rank}(G) + \lceil \log_2(2/\epsilon) \rceil$ (a bound based on the group rank) or if $k \ge \operatorname{len}(G) + \lceil \log_2(1/\epsilon) \rceil$ (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 $k \ge \lceil \log_2 |G| + \log_2(1/\epsilon) \rceil + 2$. 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.