# Non-Abelian braiding of Majorana-like edge states and topological quantum computations in electric circuits

## TL;DR

This paper demonstrates how electric circuits can simulate Majorana fermions and their non-Abelian braiding, enabling topological quantum computations through impedance measurements and circuit design.

## Contribution

It introduces a method to emulate Majorana fermions and braiding operations in electric circuits, providing a new platform for topological quantum computation simulation.

## Key findings

- Topological edge states can be observed via impedance peaks.
- Braiding operations produce quantum gates such as Clifford gates.
- Electric circuits can simulate Majorana-based topological quantum computation.

## Abstract

Majorana fermions subject to the non-Abelian braid group are believed to be the basic ingredients of future topological quantum computations. In this work, we propose to simulate Majorana fermions of the Kitaev model in electric circuits based on the observation that the circuit Laplacian can be made identical to the Hamiltonian. A set of AC voltages along the chain plays a role of the wave function. We generate an arbitrary number of topological segments in a Kitaev chain. A pair of topological edge states emerge at the edges of a topological segment. Its wave function is observable by the position and the phase of a peak in impedance measurement. It is possible to braid any pair of neighboring edge states with the aid of T-junction geometry. By calculating the Berry phase acquired by their eigenfunctions, the braiding is shown to generate one-qubit and two-qubit unitary operations. We explicitly construct Clifford quantum gates based on them. We also present an operator formalism by regarding a topological edge state as a topological soliton intertwining the trivial segment and the topological segment. Our analysis shows that the electric-circuit approach can simulate the Majorana-fermion approach to topological quantum computations.