Probabilistic Bounds on the Number of Elements to Generate Finite Nilpotent Groups and Their Applications
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909922307866624 |
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| author | Dong, Ziyuan Fan, Xiang Zhong, Tengxun Qiu, Daowen |
| author_facet | Dong, Ziyuan Fan, Xiang Zhong, Tengxun Qiu, Daowen |
| contents | This work establishes a new probabilistic bound on the number of elements 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 $φ_k(G) \ge 1 - ε$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Probabilistic Bounds on the Number of Elements to Generate Finite Nilpotent Groups and Their Applications Dong, Ziyuan Fan, Xiang Zhong, Tengxun Qiu, Daowen Quantum Physics Group Theory This work establishes a new probabilistic bound on the number of elements 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 $φ_k(G) \ge 1 - ε$ 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. |
| title | Probabilistic Bounds on the Number of Elements to Generate Finite Nilpotent Groups and Their Applications |
| topic | Quantum Physics Group Theory |
| url | https://arxiv.org/abs/2511.19494 |