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Quantum Foundations

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

arXiv

Authors: Ziyuan Dong, Xiang Fan, Tengxun Zhong, Daowen Qiu

Year

2025

Paper ID

16755

Status

Preprint

Abstract Read

~2 min

Abstract Words

143

Citations

N/A

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 log₂(2/ε) rceil (a bound based on the group rank) or if k ge operatorname{len}(G) + lceil log₂(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 log₂ |G| + log₂(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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This paper contributes to the Quantum Foundations research area in the Quantum Articles archive.
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This work establishes a new probabilistic bound on the number of elements to generate finite nilpotent groups.

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References & Citation Signals

[1] DOI https://doi.org/arXiv:2511.19494 [2] arXiv https://arxiv.org/abs/2511.19494 [3] Publisher https://arxiv.org/abs/2511.19494

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