Abstract
Braiding Majorana zero-modes around each other is a promising route toward topological quantum computing. Yet, two competing maxims emerge when implementing Majorana braiding in real systems: on the one hand, perfect braiding should be conducted adiabatically slowly to avoid non-topological errors. On the other hand, braiding must be conducted fast such that decoherence effects introduced by the environment are negligible, which are generally unavoidable in finite-size systems. This competition results in an intermediate time scale for Majorana braiding that is optimal, but generally not error-free. Here, we calculate this intermediate time scale for a T-junction of short one-dimensional topological superconductors coupled to a bosonic bath that generates fluctuations in the local electric potential, which stem from, e.g. environmental photons or phonons of the substrate. We thereby obtain boundaries for the speed of Majorana braiding with a predetermined gate fidelity. Our results emphasize the general susceptibility of Majorana-based information storage in finite-size systems and can serve as a guide for determining the optimal braiding times in future experiments.
Highlights
Majorana zero-modes are half-fermionic non-Abelian anyons [1], which have supposedly been detected in semiconducting wires in proximity to superconductors [2, 3, 4, 5], in atomic chains of transition metals on elementary superconductors [6, 7, 8], and in superconducting vortices on special systems, e.g., Fe-based superconductors [9, 10, 11, 12, 13, 14]
The boundaries for the braiding time – defined by the competition of nonadiabatic braiding errors and thermal quasiparticle excitations – are an orientation for the experimental milestone of braiding Majorana zero-modes for the first time. This procedure is important for eventual technological applications of Majorana braiding for Majorana-based quantum computing, where Γ should be well below 10−6, such that quantum error correction schemes can be applied. The structure of this manuscript is as follows: In Sec. 2, we introduce the model for the one-dimensional topological superconductors that are coupled to a bosonic bath
On the other hand, minimizing the braiding time is necessary in order to reduce errors introduced by the environment. This competition leads to an intermediate time scale for each specific system that is optimal for Majorana braiding
Summary
Majorana zero-modes are half-fermionic non-Abelian anyons [1], which have supposedly been detected in semiconducting wires in proximity to superconductors [2, 3, 4, 5], in atomic chains of transition metals on elementary superconductors [6, 7, 8], and in superconducting vortices on special systems, e.g., Fe-based superconductors [9, 10, 11, 12, 13, 14]. We first numerically simulate the finite-time braiding of Majorana zero-modes with a T-junction of short Kitaev chains [1] whose Majorana zeromodes are controlled by local gate potentials [19, 37, 36, 31], and find the minimal times where the probability for quasiparticle excitations Γ, called quasiparticle poisoning, is below a given threshold. For this we consider several threshold values between. Since the Hamiltonian in Eq (9) conserves fermionic parity, it is convenient to choose pure parity states, |± , as equivalent to the degenerate ground state |0 , |1 , where the sign ± refers to the ground state of even and odd parity, respectively
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
