Making a circulant 2-qubit entangling gate

TL;DR

This paper proposes a method to physically realize a specific circulant 2-qubit entangling gate within topological quantum computing using SU(2) Chern-Simons theory, involving braiding, fusion, and measurements.

Contribution

It introduces a novel approach to implement a circulant 2-qubit entangling gate using topological anyons, braids, and interferometric measurements in SU(2) Chern-Simons theory at level 4.

Findings

Successfully generates the circulant gate CEG in topological quantum computing.

Demonstrates the use of qubit and qutrit ancillas, braids, and measurements for gate realization.

Provides explicit matrix form of the entangling gate.

Abstract

We present a way to physically realize a circulant 2-qubit entangling gate in the Kauffman-Jones version of SU(2) Chern-Simons theory at level 4. Our approach uses qubit and qutrit ancillas, braids, fusions and interferometric measurements. Our qubit is formed by four anyons of topological charges 1221. Among other 2-qubit entangling gates we generate in the present paper, we produce in particular the circulant gate CEG = 1/4 I + I sqrt(3)/4 J - 3/4 J^2 + I sqrt(3)/4 J^3, where J denotes the permutation matrix associated with the cycle (1432) and I denotes the identity matrix.