  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 the quantum computer, 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.
