  Unitary braiding operators can be used as robust entangling quantum gates. We
introduce a solution-generating technique to solve the $(d,m,l)$-generalized
Yang-Baxter equation, for $m/2\leq l \leq m$, which allows to systematically
construct such braiding operators. This is achieved by using partition
algebras, a generalization of the Temperley-Lieb algebra encountered in
statistical mechanics. We obtain families of unitary and non-unitary braiding
operators that generate the full braid group. Explicit examples are given for a
2-, 3-, and 4-qubit system, including the classification of the entangled
states generated by these operators based on Stochastic Local Operations and
Classical Communication.
