Quantum Algorithms for Minimum Generating Set

2026-09-30

AI Summary

This study addresses the computational challenge of determining minimum generating sets for black-box groups, a problem where classical algorithms encounter significant efficiency bottlenecks, particularly for solvable groups and Γ_d-class groups containing specific non-abelian composition factors. To overcome these limitations, this work proposes polynomial-time quantum algorithms that achieve efficient computation by constructing chief series and quotient factor groups, integrated with constructive membership testing techniques. The primary contribution is the first polynomial-time quantum algorithm for Γ_d-class black-box groups, enabling the efficient determination of minimum generating sets for both solvable groups and this broader class. Furthermore, it establishes that the general problem resides within the NP∩coAM complexity class, thereby transcending the efficiency constraints inherent in traditional classical approaches.

Abstract

In this paper, we present a polynomial-time quantum algorithm for computing a minimum-sized generating set of solvable black-box groups. Next, we consider the class $Γ_d$ of black-box groups, where every non-abelian composition factor is isomorphic to a subgroup of the symmetric group $S_d$ for a fixed $d$. We design polynomial-time quantum algorithms to compute the direct product decomposition of abelian factor groups and solve the constructive membership problem for factor groups of groups from $Γ_d$. With the help of these algorithms, we design a quantum algorithm for computing a chief series of black-box groups from $Γ_d$. Using the chief series, we construct a polynomial-time quantum algorithm for computing minimum generating sets of black-box groups from $Γ_d$. Finally, we show that the minimum generating set problem for general black-box groups is in $\textrm{NP} \cap \textrm{coAM}$.

Problem

• minimum generating set
• black-box groups
• solvable groups
• chief series
• computational complexity

Innovation

• Quantum Algorithms
• Minimum Generating Set
• Black-box Groups
• Chief Series
• Computational Complexity