Some Coxeter Groups in Reversible and Quantum Compuation

TL;DR

This paper explores how Coxeter groups underpin certain gate sets in reversible and quantum computing, enabling potential circuit optimization through efficient word problems, exemplified by the 3SAT oracle.

Contribution

It demonstrates the presence of Coxeter group structures in quantum gate sets and their application to circuit optimization, particularly for complex problems like 3SAT.

Findings

Coxeter groups are present in quantum gate sets.

Efficient word problems can help shorten quantum circuits.

Application to 3SAT oracle illustrates practical benefits.

Abstract

In this article we show how the structure of Coxeter groups are present in gate sets of reversible and quantum computing. These groups have efficient word problems which means that circuits built from these gates have potential to be shortened efficiently. This is especially useful in the case of quantum computing when one does not have the timescale to perform a long series of gates and so one must and a gate scheduling that minimizes circuit depth. As the main example we consider the oracle for 3SAT.