Counting unitaries of T-depth one

Abstract

We show that the number of T-depth one unitaries on $n$ qubits is $ \sum_{m=1}^{n}\tfrac{1}{m!}\prod_{k=0}^{m-1}(4^n/2^k - 2^k) \times \# \mathcal{C}_n, $ where $\#\mathcal{C}_n$ is the size of the $n$-qubit Clifford group, that is the number of unitaries of T-depth zero. The number of T-depth one unitaries on $n$ qubits grows as $2^{\Omega(n^2)} \cdot \# \mathcal{C}_n$.