Working Paper Article Version 1 This version is not peer-reviewed

# Polyadic Braid Operators and Higher Braiding Gates

Version 1 : Received: 9 July 2021 / Approved: 12 July 2021 / Online: 12 July 2021 (09:08:06 CEST)

A peer-reviewed article of this Preprint also exists.

Duplij, S.; Vogl, R. Polyadic Braid Operators and Higher Braiding Gates. Universe 2021, 7, 301. Duplij, S.; Vogl, R. Polyadic Braid Operators and Higher Braiding Gates. Universe 2021, 7, 301.

Journal reference: Universe 2021, 7, 301
DOI: 10.3390/universe7080301

## 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 which can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in details. We also consider exotic noninvertible 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 noninvertible). 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 which are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.

## Keywords

Yang-Baxter equation; braid group; qubit; ternary; polyadic; braiding quantum gate

## Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

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