Quantum circuits of T-depth one

We give a Clifford+T representation of the Toffoli gate of T -depth 1, using four ancillas. More generally, we describe a class of circuits whose T -depth can be reduced to 1 by using sufficiently many ancillas. We show that the cost of adding an additional control to any controlled gate is at most 8 additional T -gates, and T -depth 2. We also show that the circuit THT does not possess a T -depth 1 representation with an arbitrary number of ancillas initialized to |0〉.