Probability distribution of Majorana end-state energies in disordered wires

TL;DR

This paper analyzes the statistical distribution of Majorana end-state energies in disordered one-dimensional topological superconducting wires, revealing a log-normal distribution for the energy splitting and implications for quantum computing.

Contribution

It provides a detailed characterization of the probability distribution of Majorana energy splittings in disordered wires, highlighting the effects of disorder on their fluctuations.

Findings

Energy splitting distribution is log-normal for large wires.

Bulk energy levels have algebraic tail distributions.

Disorder influences the stability and operation speed of topological quantum computers.

Abstract

One-dimensional topological superconductors harbor Majorana bound states at their ends. For superconducting wires of finite length L, these Majorana states combine into fermionic excitations with an energy $\epsilon_0$ that is exponentially small in L. Weak disorder leaves the energy splitting exponentially small, but affects its typical value and causes large sample-to-sample fluctuations. We show that the probability distribution of $\epsilon_0$ is log normal in the limit of large L, whereas the distribution of the lowest-lying bulk energy level $\epsilon_1$ has an algebraic tail at small $\epsilon_1$. Our findings have implications for the speed at which a topological quantum computer can be operated.