  Higher braiding gates, a new kind of quantum gate, are introduced. These are
matrix solutions of the polyadic braid equations (which differ from the
generalized Yang-Baxter equations). Such gates support a special kind of
multi-qubit entanglement which can speed up key distribution and accelerate the
execution of algorithms. Ternary braiding gates acting on three qubit states
are studied in detail. We also consider exotic non-invertible gates which can
be related to qubit loss, and define partial identities (which can be
orthogonal), partial unitarity, and partially bounded operators (which can be
non-invertible). We define two classes of matrices, the star and circle types,
and find that the magic matrices (connected with the Cartan decomposition)
belong to the star class. The general algebraic structure of the classes
introduced here is described in terms of semigroups, ternary and 5-ary groups
and modules. The higher braid group and its representation by higher braid
operators are given. Finally, we show that for each multi-qubit state there
exist higher braiding gates which are not entangling, and the concrete
conditions to be non-entangling are given for the binary and ternary gates
discussed.
