  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.
