Generators and Relations for the Group On(Z[1/2])

TL;DR

This paper provides a finite presentation of certain orthogonal matrix groups over Z[1/2], linking algebraic structures to quantum circuit representations, especially for dimensions that are powers of two.

Contribution

It introduces explicit generators and relations for these groups, connecting algebraic group theory with quantum circuit implementation.

Findings

Finite presentation for O_n(Z[1/2])

Explicit description of matrices representable by quantum circuits

Characterization of unitary matrices in quantum computing context

Abstract

We give a finite presentation by generators and relations for the group O_n(Z[1/2]) of n-dimensional orthogonal matrices with entries in Z[1/2]. We then obtain a similar presentation for the group of n-dimensional orthogonal matrices of the form M/sqrt(2)^k, where k is a nonnegative integer and M is an integer matrix. Both groups arise in the study of quantum circuits. In particular, when the dimension is a power of 2, the elements of the latter group are precisely the unitary matrices that can be represented by a quantum circuit over the universal gate set consisting of the Toffoli gate, the Hadamard gate, and the computational ancilla.