Paper: Generators and relations for the group of Orthogonal Dyadic operators
Check out the 4th International Workshop on Quantum Compilation (IWQC 2020).
Abstract: We give a finite presentation by generators and relations for the group On(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 (1/2)^kM, 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 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.