Quantum circuits of T-depth one

TL;DR

This paper presents a method to construct quantum circuits with T-depth 1 using ancillas, focusing on the Toffoli gate and controlled gates, and discusses limitations of certain circuit transformations.

Contribution

It introduces a Clifford+T representation of the Toffoli gate with T-depth 1 and analyzes the cost of adding controls to controlled gates.

Findings

Clifford+T representation of Toffoli with T-depth 1 using four ancillas

Adding a control increases T-gates by at most 8 and T-depth to 2

Some circuits like THT cannot achieve T-depth 1 with any number of ancillas

Abstract

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.