Both Toffoli and Controlled-NOT need little help to do universal quantum computation

TL;DR

This paper demonstrates that the combination of Toffoli and Controlled-NOT gates, along with any single-qubit real gate (except those preserving the computational basis), is sufficient for universal quantum computation, expanding understanding of gate universality.

Contribution

It proves that Controlled-NOT plus any single-qubit real gate (not preserving the computational basis) is universal for quantum computing, broadening the class of known universal gate sets.

Findings

Controlled-NOT plus any non-basis-preserving real gate is universal.

Controlled-NOT and Hadamard gates can be classically simulated efficiently.

Most single-qubit real gates, except basis-preserving ones, enable universal quantum computation.

Abstract

What additional gates are needed for a set of classical universal gates to do universal quantum computation? We answer this question by proving that any single-qubit real gate suffices, except those that preserve the computational basis. The result of Gottesman and Knill[quant-ph/9807006] implies that any quantum circuit involving only the Controlled-NOT and Hadamard gates can be efficiently simulated by a classical circuit. In contrast, we prove that Controlled-NOT plus any single-qubit real gate that does not preserve the computational basis and is not Hadamard (or its alike) are universal for quantum computing. Previously only a ``generic'' gate, namely a rotation by an angle incommensurate with pi, is known to be sufficient in both problems, if only one single-qubit gate is added.