Quantum algorithm for the hidden subgroup problem on a class of semidirect product groups

TL;DR

This paper introduces efficient quantum algorithms for solving the hidden subgroup problem on specific semidirect product groups, achieving exponential speedup over classical methods.

Contribution

It develops the first quantum algorithms for HSP on certain semidirect product groups involving cyclic groups, expanding the class of groups where quantum speedups are known.

Findings

Quantum algorithms are exponentially faster than classical algorithms for these groups.

Algorithms work for groups of the form Z_{p^r} ⋉ Z_{p^2} with odd prime p and r > 4.

Addresses HSP in groups with special prime factorizations.

Abstract

We present efficient quantum algorithms for the hidden subgroup problem (HSP) on the semidirect product of cyclic groups $\Z_{p^r}\rtimes_{\phi}\Z_{p^2}$, where $p$ is any odd prime number and $r$ is any integer such that $r>4$. We also address the HSP in the group $\Z_{N}\rtimes_{\phi}\Z_{p^2}$, where $N$ is an integer with a special prime factorization. These quantum algorithms are exponentially faster than any classical algorithm for the same purpose.