Application of quantum algorithms to the study of permutations and group automorphisms

TL;DR

This paper explores quantum algorithms for analyzing permutations and group automorphisms, demonstrating how quantum techniques can efficiently determine properties like fixed points, cycles, and automorphism mappings, with implications for quantum information processing.

Contribution

It introduces novel quantum algorithms for permutation and automorphism analysis, including implementations via programmable quantum processors, advancing quantum information processing capabilities.

Findings

Quantum algorithms can determine automorphism properties efficiently.

Programmable quantum processors can implement these algorithms.

New methods for analyzing group automorphisms using quantum computing.

Abstract

We discuss three applications of efficient quantum algorithms to determining properties of permutations and group automorphisms. The first uses the Bernstein-Vazirani algorithm to determine an unknown homomorphism from $Z_{p-1}^{m}$ to $Aut(Z_{p})$ where $p$ is prime. The remaining two make use of modifications of the Grover search algorithm. The first finds the fixed point of a permutation or an automorphism (assuming it has only one besides the identity). It can be generalized to find cycles of a specified size for permutations or orbits of a specified size for automorphisms. The second finds which of a set of permutations or automorphisms maps one particular element of a set or group onto another. This has relevance to the conjugacy problem for groups. We show how two of these algorithms can be implemented via programmable quantum processors. This approach opens new perspectives in quantum information processing, wherein both the data and the programs are represented by states of quantum registers. In particular, quantum programs that specify control over data can be treated using methods of quantum information theory.