Generators and Relations for the Group On(Z[1/2])
Quantum Physics
2021-09-14 v2 Emerging Technologies
Logic in Computer Science
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.
Cite
@article{arxiv.2106.01175,
title = {Generators and Relations for the Group On(Z[1/2])},
author = {Sarah Meng Li and Neil J. Ross and Peter Selinger},
journal= {arXiv preprint arXiv:2106.01175},
year = {2021}
}
Comments
In Proceedings QPL 2021, arXiv:2109.04886