Quantum representation of finite groups

TL;DR

This paper formally defines quantum representations of finite groups, proves their universal existence, and demonstrates their application in quantum algorithms and automata through numerical simulations.

Contribution

It introduces a formal definition of QRFG, proves their existence for all finite groups, and provides methods for quantum circuit implementation.

Findings

Proved existence of quantum representations for all finite groups.

Presented two methods for translating group generators into quantum circuits.

Demonstrated applications in quantum algorithms and automata with numerical simulations.

Abstract

The concept of quantum representation of finite groups (QRFG) has been a fundamental aspect of quantum computing for quite some time, playing a role in every corner, from elementary quantum logic gates to the famous Shor's and Grover's algorithms. In this article, we provide a formal definition of this concept using both group theory and differential geometry. Our work proves the existence of a quantum representation for any finite group and outlines two methods for translating each generator of the group into a quantum circuit, utilizing gate decomposition of unitary matrices and variational quantum algorithms. Additionally, we provide numerical simulations of an explicit example on an open-access platform. Finally, we demonstrate the usefulness and potential of QRFG by showing its role in the implementation of some quantum algorithms and quantum finite automata.