Braiding with Majorana lattices: Groundstate degeneracy and supersymmetry

TL;DR

This paper proposes a braiding protocol for Majorana lattices in topological superconductors, demonstrating that groundstate degeneracy improves exponentially with system size due to supersymmetry and topology, aiding fault-tolerant quantum computation.

Contribution

It introduces a universal braiding scheme for Majorana lattices that leverages supersymmetry and topology to reduce energy splitting, enhancing qubit stability.

Findings

Energy splitting decreases exponentially with the number of Majorana modes.

The result is independent of specific braiding geometry and scheme details.

Supersymmetry and topology underpin the degeneracy preservation.

Abstract

Majorana-based topological qubits are expected to exploit the nonabelian braiding statistics of Majorana modes in topological superconductors to realize fault-tolerant topological quantum computation. Scalable qubit designs require several Majorana modes localized on quantum wires networks, with braiding operations relying on the presence of the groundstate degeneracy of the topologically nontrivial superconducting phase. However, this degeneracy is lifted due to the hybridization between Majorana modes localized at a finite distance. Here, we describe a braiding protocol in a trijunction where each branch consists of a lattice of Majorana modes overlapping at a finite distance. We find that the energy splitting between the groundstate and the lowest-energy state decreases exponentially with the number of Majorana modes if the system is in its topologically nontrivial regime. This result does not rely on the specific braiding geometry and on the details of the braiding scheme but is a consequence of the supersymmetry and nontrivial topology of the effective low-energy Hamiltonian describing the Majorana lattice.