Polyadic braid operators and higher braiding gates

TL;DR

This paper introduces higher braiding gates as solutions to polyadic braid equations, exploring their algebraic structures, entangling capabilities, and potential applications in quantum information processing.

Contribution

It presents the concept of higher braiding gates, analyzes their algebraic properties, and characterizes conditions for entanglement and non-entanglement in multi-qubit systems.

Findings

Higher braiding gates support special multi-qubit entanglement.

Conditions for non-entangling gates are explicitly derived.

The algebraic structure involves semigroups, ternary and 5-ary groups.

Abstract

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.