On group theory for quantum gates and quantum coherence

TL;DR

This paper applies finite group theory to quantum computing, analyzing the structure of Clifford groups and revealing the role of the M20 group in quantum coherence for two-qubit systems.

Contribution

It explores the connection between group theoretical concepts and quantum computational primitives, providing detailed analysis of Clifford groups and identifying the M20 group as fundamental to quantum coherence.

Findings

Clifford groups' structure is analyzed in detail.

M20 group underpins quantum coherence in two-qubit systems.

Automorphisms of mutually unbiased bases relate to group structures.

Abstract

Finite group extensions offer a natural language to quantum computing. In a nutshell, one roughly describes the action of a quantum computer as consisting of two finite groups of gates: error gates from the general Pauli group P and stabilizing gates within an extension group C. In this paper one explores the nice adequacy between group theoretical concepts such as commutators, normal subgroups, group of automorphisms, short exact sequences, wreath products... and the coherent quantum computational primitives. The structure of the single qubit and two-qubit Clifford groups is analyzed in detail. As a byproduct, one discovers that M20, the smallest perfect group for which the commutator subgroup departs from the set of commutators, underlies quantum coherence of the two-qubit system. One recovers similar results by looking at the automorphisms of a complete set of mutually unbiased bases.