@article{Dong2026, 
author = {Ziyuan Dong and Xiang Fan and Tengxun Zhong and Daowen Qiu},
title = {Probabilistic bounds on the number of elements to generate finite nilpotent groups and their applications to quantum algorithms},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {4},
pages = {9380-9397},
keywords = {probabilistic group theory, group generation, finite nilpotent group, abelian hidden subgroup problem, quantum algorithms},
url = {https://www.sciopen.com/article/10.3934/math.2026389},
doi = {10.3934/math.2026389},
abstract = {This work establishes a new probabilistic bound on the number of elements needed 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  )  ≥  1  −  ϵ if    k  ≥  rank  ⁡  (  G  )  +  ⌈      log    2    ⁡  (  2      /    ϵ  )  ⌉ (a bound based on the group rank) or if    k  ≥  len  ⁡  (  G  )  +  ⌈      log    2    ⁡  (  1      /    ϵ  )  ⌉ (a bound based on the composition length). Moreover, these bounds are shown to be nearly tight. Both bounds sharpen the previously known requirement of    k  ≥  ⌈      log    2    ⁡      |    G      |    +      log    2    ⁡  (  1      /    ϵ  )  ⌉  +  2. Our results provide a foundational tool to analyze probabilistic algorithms, thereby 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.}
}