Quantum Threshold is Powerful

TL;DR

This paper demonstrates that quantum Threshold gates are powerful enough to simulate Fanout gates within constant-depth circuits, expanding understanding of quantum gate capabilities.

Contribution

It proves that quantum Threshold gates can replicate Fanout gates in constant-depth circuits, generalizing previous results and revealing new gate equivalences.

Findings

Threshold gates can simulate Fanout with high fidelity

Other gates approximating Parity can substitute Fanout

Constant-depth quantum circuits with Threshold gates are powerful

Abstract

In 2005, H{\o}yer and \v{S}palek showed that constant-depth quantum circuits augmented with multi-qubit Fanout gates are quite powerful, able to compute a wide variety of Boolean functions as well as the quantum Fourier transform. They also asked what other multi-qubit gates could rival Fanout in terms of computational power, and suggested that the quantum Threshold gate might be one such candidate. Threshold is the gate that indicates if the Hamming weight of a classical basis state input is greater than some target value. We prove that Threshold is indeed powerful--there are polynomial-size constant-depth quantum circuits with Threshold gates that compute Fanout to high fidelity. Our proof is a generalization of a proof by Rosenthal that exponential-size constant-depth circuits with generalized Toffoli gates can compute Fanout. Our construction reveals that other quantum gates able to "weakly approximate" Parity can also be used as substitutes for Fanout.