# Error generation and propagation in Majorana-based topological qubits

## TL;DR

This paper analyzes error mechanisms in Majorana-based topological qubits, focusing on qubit-loss and secondary errors due to non-adiabaticity, and explores how system parameters influence error rates and mitigation strategies.

## Contribution

It provides a detailed study of error generation and propagation in topological qubits, including effects of boundary driving, movement of Majorana states, and disorder, offering insights into error minimization.

## Key findings

- Qubit-loss rate correlates with local density of states at wire edges.
- Super-adiabaticity and critical velocity do not reduce qubit-loss during Majorana shuttling.
- Disordering central regions of wires can minimize certain error channels.

## Abstract

We investigate dynamical evolution of a topological memory that consists of two p-wave superconducting wires separated by a non-topological junction, focusing on the primary errors (i.e., qubit-loss) and secondary errors (bit and phase-flip) that arise due to non-adiabaticity. On the question of qubit-loss we examine the system's response to both periodic boundary driving and deliberate shuttling of the Majorana bound states. In the former scenario we show how the frequency dependent rate of qubit-loss is strongly correlated with the local density of states at the edge of wire, a fact that can make systems with a larger gap more susceptible to high frequency noise. In the second scenario we confirm previous predictions concerning super-adiabaticity and critical velocity, but see no evidence that the coordinated movement of edge boundaries reduces qubit-loss. Our analysis on secondary bit flip errors shows that it is necessary that non-adiabaticity occurs in both wires and that inter-wire tunnelling be present for this error channel to be open. We also demonstrate how such processes can be minimised by disordering central regions of both wires. Finally we identify an error channel for phase flip errors, which can occur due to mismatches in the energies of states with bulk excitations. In the non-interacting system considered here this error systematically opposes the expected phase rotation due to finite size splitting in the qubit subspace.