Polyadic Braid Operators and Higher Braiding Gates

Abstract

A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang--Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with 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, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and $5$-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.Yang--Baxter equation; braid group; qubit; ternary; polyadic; braiding quantum gate.

Polyadization of Algebraic Structures

“…Notice that the P-matrices (35) are the block-matrix versions of the circle matrices M circ , which were studied in [32] in connection with 8-vertex solutions to the constant Yang-Baxter equation [33] and the corresponding braiding quantum gates [34,35].…”