Unitary tetrahedron quantum gates

TL;DR

This paper systematically constructs and classifies unitary tetrahedron (3-simplex) quantum gates using Clifford algebras and Yang-Baxter operator lifting, enabling new multi-qubit gates for quantum computing.

Contribution

It provides the first comprehensive classification of unitary tetrahedron operators for qubits, expanding the toolkit for quantum circuit design.

Findings

13 inequivalent families of unitary tetrahedron operators identified

12 families derived from known Yang-Baxter operators and one from Clifford algebra

Universal sets of 1-, 2-, and 3-qubit gates constructed from these operators

Abstract

Quantum simulations of many-body systems using 2-qubit Yang-Baxter gates offer a benchmark for quantum hardware. This can be extended to the higher dimensional case with $n$-qubit generalisations of Yang-Baxter gates called $n$-simplex operators. Such multi-qubit gates potentially lead to shallower and more efficient quantum circuits as well. Finding them amounts to identifying unitary solutions of the $n$-simplex equations, the building blocks of higher dimensional integrable systems. These are a set of highly non-linear and over determined system of equations making it notoriously hard to solve even when the local Hilbert spaces are spanned by qubits. We systematically overcome this for higher simplex operators constructed using two methods: from Clifford algebras and by lifting Yang-Baxter operators. The $n=3$ or the tetrahedron case is analyzed in detail. For the qubit case our methods produce 13 inequivalent families of unitary tetrahedron operators. 12 of these families are obtained by appending the 5 unitary families of 4 by 4 constant Yang-Baxter operators of Dye-Hietarinta, with a single qubit operator. As applications, universal sets of single, two and three qubit gates are realized using such unitary tetrahedron operators. The ideas presented in this work can be naturally extended to the higher simplex cases.