Quantum Circuits for Functionally Controlled NOT Gates

TL;DR

This paper develops quantum circuit constructions for functionally controlled NOT gates, extending previous work on Toffoli gates, using Walsh-Hadamard spectra and optimizing for auxiliary qubits and rotation depth.

Contribution

It introduces new quantum circuit designs for functionally controlled NOT gates with minimal auxiliary qubits and rotation depth, generalizing prior Toffoli gate implementations.

Findings

Circuits for arbitrary target states

No auxiliary qubits required in some constructions

Rotation depth of 1 achieved in certain cases

Abstract

We generalize quantum circuits for the Toffoli gate presented by Selinger and Jones for functionally controlled NOT gates, i.e., $X$ gates controlled by arbitrary $n$-variable Boolean functions. Our constructions target the gate set consisting of Clifford gates and single qubit rotations by arbitrary angles. Our constructions use the Walsh-Hadamard spectrum of Boolean functions and build on the work by Schuch and Siewert and Welch et al. We present quantum circuits for the case where the target qubit is in an arbitrary state as well as the special case where the target is in a known state. Additionally, we present constructions that require no auxiliary qubits and constructions that have a rotation depth of 1.