Probabilistic bounds on the number of elements to generate finite nilpotent groups and their applications to quantum algorithms

This work establishes a new probabilistic bound on the number of elements needed to generate finite nilpotent groups. Let ${\phi }_k(G)$ denote the probability that $k$ random elements generate a finite nilpotent group $G$. For any $0<ϵ<1$, we prove that ${\phi }_k(G)\ge 1-ϵ$ if $k\ge \mathrm{rank}(G)+\lceil {\log }_2(2/ϵ)\rceil$ (a bound based on the group rank) or if $k\ge \mathrm{len}(G)+\lceil {\log }_2(1/ϵ)\rceil$ (a bound based on the composition length). Moreover, these bounds are shown to be nearly tight. Both bounds sharpen the previously known requirement of $k\ge \lceil {\log }_2|G|+{\log }_2(1/ϵ)\rceil +2$. 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.