Quantum circuits of T-depth one

We give a Clifford + T representation of the Toffoli gate of T-depth one, using four ancillas. More generally, we describe a class of circuits whose T-depth can be reduced to one by using sufficiently many ancillas. We show that the cost of adding an additional control to any controlled gate is at most eight additional T gates and T-depth two. We also show that the circuit THT does not possess a T-depth one representation with an arbitrary number of ancillas initialized to vertical bar 0 >. DOI: 10.1103/PhysRevA.87.042302