Extraspecial Two-Groups, Generalized Yang-Baxter Equations and Braiding Quantum Gates

TL;DR

This paper explores new connections between extraspecial 2-groups, braid group representations, and quantum gates, leading to novel multi-qubit braiding gates capable of generating GHZ states, with implications for quantum error correction and topological quantum computing.

Contribution

It introduces new unitary braid group representations derived from extraspecial 2-groups and extends them to construct braiding quantum gates for multi-qubit systems.

Findings

New representations of extraspecial 2-groups constructed.

Unitary braid representations solving generalized Yang-Baxter equations developed.

Braid gates capable of generating GHZ states for arbitrary qubit numbers.

Abstract

In this paper we describe connections among extraspecial 2-groups, unitary representations of the braid group and multi-qubit braiding quantum gates. We first construct new representations of extraspecial 2-groups. Extending the latter by the symmetric group, we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates generate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly an \emph{odd}) number of qubits, from the product basis. We also discuss the Yang-Baxterization of the new braid group representations, which describes unitary evolution of the GHZ states. Our study suggests that through their connection with braiding gates, extraspecial 2-groups and the GHZ states may play an important role in quantum error correction and topological quantum computing.