Quantum Algorithms for Minimum Generating Set

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 $\Gamma_d$ 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 $\Gamma_d$. With the help of these algorithms, we design a quantum algorithm for computing a chief series of black-box groups from $\Gamma_d$. Using the chief series, we construct a polynomial-time quantum algorithm for computing minimum generating sets of black-box groups from $\Gamma_d$. Finally, we show that the minimum generating set problem for general black-box groups is in $\textrm{NP} \cap \textrm{coAM}$.

Quantum Physics (quant-ph); Computational Complexity (cs.CC); Group Theory (math.GR)

arXiv:2609.40260 [quant-ph]

(or arXiv:2609.40260v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.40260 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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