Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors

TL;DR

This paper investigates non-adiabatic effects on the braiding of Majorana fermions in topological superconductors, revealing conditions under which non-Abelian statistics are preserved or disrupted due to system state localization.

Contribution

It introduces a formalism using time-dependent Bogoliubov-de Gennes equations to analyze non-adiabatic corrections in Majorana braiding, highlighting the impact of state localization on non-Abelian statistics.

Findings

Localized bound states allow recovery of non-Abelian statistics with proper fermion parity definition

Extended states forming a continuum disrupt local fermion parity and non-Abelian statistics

Quantitative characterization of errors due to non-adiabatic effects in different state configurations

Abstract

Qubits in topological quantum computation are built from non-Abelian anyons. Adiabatic braiding of anyons is exploited as topologically protected logical gate operations. Thus, the adiabaticity upon which the notion of quantum statistics is defined, plays a fundamental role in defining the non-Abelian anyons. We study the non-adiabatic effects in braidings of Ising-type anyons, namely Majorana fermions in topological superconductors, using the formalism of time-dependent Bogoliubov-de Gennes equations. Using this formalism, we consider non-adiabatic corrections to non-Abelian statistics from: (1) tunneling splitting of anyons imposing an additional dynamical phase to the transformation of ground states; (2) transitions to excited states that are potentially destructive to non-Abelian statistics since the non-local fermion occupation can be spoiled by such processes. However, if the bound states are localized and being braided together with the anyons, non-Abelian statistics can be recovered once the definition of Majorana operators is appropriately generalized taking into account the fermion parity in these states. On the other hand, if the excited states are extended over the whole system and form a continuum, the notion of local fermion parity no longer holds. We then quantitatively characterize the errors introduced in this situation.