Generators and Relations for Un(Z[1/2,i])

TL;DR

This paper provides a finite presentation of the group of unitary matrices with entries in Z[1/2,i], expanding understanding of quantum gate sets with entries in this specific ring.

Contribution

It offers a finite generators-and-relations presentation for U_n(Z[1/2,i]), the group of unitary matrices over Z[1/2,i], complementing prior work on circuit realizations.

Findings

Finite presentation of U_n(Z[1/2,i]) established

Connections between algebraic structures and quantum gate sets clarified

Enhances understanding of gates with entries in Z[1/2,i]

Abstract

Consider the universal gate set for quantum computing consisting of the gates X, CX, CCX, omega^dagger H, and S. All of these gates have matrix entries in the ring Z[1/2,i], the smallest subring of the complex numbers containing 1/2 and i. Amy, Glaudell, and Ross proved the converse, i.e., any unitary matrix with entries in Z[1/2,i] can be realized by a quantum circuit over the above gate set using at most one ancilla. In this paper, we give a finite presentation by generators and relations of U_n(Z[1/2,i]), the group of unitary nxn-matrices with entries in Z[1/2,i].