Polyadic braid operators and higher braiding gates

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. CONTENTS I. Introduction II. Yang-Baxter operators A. Yang-Baxter maps and braid group B. Constant matrix solutions to the Yang-Baxter equation C. Partial identity and unitarity D. Permutation and parameter-permutation 4-vertex Yang-Baxter maps E. Group structure of 4-vertex and 8-vertex matrices F. Star 8-vertex and circle 8-vertex Yang-Baxter maps G. Triangle invertible 9and 10-vertex solutions III. Polyadic braid operators and higher braid equations IV. Solutions to the ternary braid equations A. Constant matrix solutions B. Permutation and parameter-permutation 8-vertex solutions C. Group structure of the star and circle 8-vertex matrices D. Group structure of the star and circle 16-vertex matrices E. Pauli matrix presentation of the star and circle 16-vertex constant matrices F. Invertible and non-invertible 16-vertex solutions to the ternary braid equations G. Higher 2 n -vertex constant solutions to n-ary braid equations V. Invertible and noninvertible quantum gates VI. Binary braiding quantum gates VII. Higher braiding quantum gates VIII. Entangling braiding gates A. Entangling binary braiding gates B. Entangling ternary braiding gates References