Geometric representations of braid and Yang-Baxter gates

TL;DR

This paper explores the geometric representations of braid and Yang-Baxter gates, identifying their properties, optimal decompositions, and conditions for implementation on quantum computers, which advances quantum circuit design and understanding.

Contribution

It provides a comprehensive geometric framework for braid and Yang-Baxter gates, including their classification, optimal decompositions, and implementation conditions on quantum hardware.

Findings

Braid and Yang-Baxter gates are confined to specific regions of a two-qubit tetrahedron.

Identified parameters where these gates are Clifford, matchgate, or dual-unitary.

Entangling power of Yang-Baxter gates depends on spectral parameters.

Abstract

Brick-wall circuits composed of the Yang-Baxter gates are integrable. It becomes an important tool to study the quantum many-body system out of equilibrium. To put the Yang-Baxter gate on quantum computers, it has to be decomposed into the native gates of quantum computers. It is favorable to apply the least number of native two-qubit gates to construct the Yang-Baxter gate. We study the geometric representations of all X-type braid gates and their corresponding Yang-Baxter gates via the Yang-Baxterization. We find that the braid and Yang-Baxter gates can only exist on certain edges and faces of the two-qubit tetrahedron. We identify the parameters by which the braid and Yang-Baxter gates are the Clifford gate, the matchgate, and the dual-unitary gate. The geometric representations provide the optimal decompositions of the braid and Yang-Baxter gates in terms of other two-qubit gates. We also find that the entangling powers of the Yang-Baxter gates are determined by the spectral parameters. Our results provide the necessary conditions to construct the braid and Yang-Baxter gates on quantum computers.