Toffoli gates solve the tetrahedron equations

TL;DR

This paper demonstrates that Toffoli gates, essential for universal quantum computation, can be constructed using 3-simplex operators satisfying tetrahedron equations, extending integrable models to higher qubit gates.

Contribution

It introduces a novel connection between Toffoli gates and 3-simplex operators satisfying tetrahedron equations, expanding the framework of integrable quantum circuits.

Findings

Toffoli gates can be realized via 3-simplex operators.

These operators satisfy a spectral parameter-dependent tetrahedron equation.

The construction extends to n-Toffoli gates with n-simplex operators.

Abstract

The circuit model of quantum computation can be interpreted as a scattering process. In particular, factorised scattering operators result in integrable quantum circuits that provide universal quantum computation and are potentially less noisy. These are realized through Yang-Baxter or 2-simplex operators. A natural question is to extend this construction to higher qubit gates, like the Toffoli gates, which also lead to universal quantum computation but with shallower circuits. We show that unitary families of such operators are constructed by the 3-dimensional generalizations of the Yang-Baxter operators known as tetrahedron or 3-simplex operators. The latter satisfy a spectral parameter-dependent tetrahedron equation. This construction goes through for $n$-Toffoli gates realized using $n$-simplex operators.