A Simple Proof that Toffoli and Hadamard are Quantum Universal

TL;DR

This paper provides a straightforward proof that the Toffoli and Hadamard gates form a universal set for quantum computation, simplifying the understanding of quantum gate universality.

Contribution

It offers a concise proof of Toffoli and Hadamard's universality, building on Kitaev's gate set, and clarifies the significance of this result.

Findings

Toffoli and Hadamard gates are universal for quantum computation.

The proof is simple and relies on Kitaev's universal gate set.

The result emphasizes the conceptual simplicity of quantum universality.

Abstract

Recently Shi proved that Toffoli and Hadamard are universal for quantum computation. This is perhaps the simplest universal set of gates that one can hope for, conceptually; It shows that one only needs to add the Hadamard gate to make a 'classical' set of gates quantum universal. In this note we give a few lines proof of this fact relying on Kitaev's universal set of gates, and discuss the meaning of the result.