Any Clifford+T circuit can be controlled with constant T-depth overhead

TL;DR

This paper demonstrates that any Clifford+T quantum circuit can be controlled with constant T-depth overhead, significantly improving control efficiency without ancillas or measurement, and introduces applications like precise rotation catalysis.

Contribution

It introduces methods to control Clifford+T circuits with constant T-depth overhead, reducing complexity and enabling new quantum control applications.

Findings

Controlled Clifford circuits can be implemented with O(1) T-depth.

T-depth of controlled Clifford+T circuits scales linearly with original T-depth.

Exact rotation catalysis achieved with T-depth 1 using a small catalyst state.

Abstract

Since an n-qubit circuit consisting of CNOT gates can have up to $\Omega(n^2/\log{n})$ CNOT gates, it is natural to expect that $\Omega(n^2/\log{n})$ Toffoli gates are needed to apply a controlled version of such a circuit. We show that the Toffoli count can be reduced to at most n. The Toffoli depth can also be reduced to O(1), at the cost of 2n Toffoli gates, even without using any ancilla or measurement. In fact, using a measurement-based uncomputation, the Toffoli depth can be further reduced to 1. From this, we give two corollaries: any controlled Clifford circuit can be implemented with O(1) T-depth, and any Clifford+T circuit with T-depth D can be controlled with T-depth O(D), even without ancillas. As an application, we show how to catalyze a rotation by any angle up to precision $\epsilon$ in T-depth exactly 1 using a universal $\lceil\log_2(8/\epsilon)\rceil$-qubit catalyst state.