Topological gap protocol based machine learning optimization of Majorana hybrid wires

TL;DR

This work tackles disorder-induced destruction of the topological phase in Majorana hybrid wires by optimizing a near-wire gate array using the CMA-ES algorithm. A topological-gap-based metric, computable from conductance measurements, guides the optimization to restore localized Majorana zero modes and a finite excitation gap without requiring interferometry. The approach successfully compensates strong disorder in both one- and two-dimensional wires, can ignore trivial Andreev bound states, and converges with a practical number of metric evaluations. The results offer a scalable, experimentally feasible route to robust Majorana qubits and motivate gate-based tuning as a core tool for disorder resilience in topological quantum devices.

Abstract

Majorana zero modes in superconductor-nanowire hybrid structures are a promising candidate for topologically protected qubits with the potential to be used in scalable structures. Currently, disorder in such Majorana wires is a major challenge, as it can destroy the topological phase and thus reduce the yield in the fabrication of Majorana devices. We study machine learning optimization of a gate array in proximity to a grounded Majorana wire, which allows us to reliably compensate even strong disorder. We propose a metric for optimization that is inspired by the topological gap protocol, and which can be implemented based on measurements of the non-local conductance through the wire.

Figures (11)

• Figure 1: Majorana hybrid wire consisting of a grounded superconductor (orange) and a semiconductor with strong spin orbit coupling (blue) connected to two leads $L$ and $R$ separated from the wire by a potential $V_{\rm conf}$ created by pinch-off gates. The full conductance matrix $G_{\alpha\beta}=\mathrm{d} I_\alpha/\mathrm{d} V_{\beta}$ can be measured as a function of an applied bias voltage $V_R-V_L$ and external Zeeman field $E_z$ based on which voltages of an array of gates (green) are optimized using the CMA-ES algorithm Hansen.2016 to cancel disorder effects in the hybrid wire.
• Figure 2: Effects of strong disorder on a Majorana wire of length $L=32.5\,l_{\rm so}$ in the topological phase for $\mu=1\,E_{\rm so}$ and $E_z=6\,E_{\rm so}$. Panels (a) and (b) depict the wave functions $|\Psi_0|^2$ (blue) of the ground state together with the electron $|u_0|^2$ (green) and hole wave function $|v_0|^2$ (orange) for a clean and disordered wire ($\sigma_{\rm dis}=25\,E_{\rm so}$, $\lambda_{\rm dis}=0$), respectively. Panels (c) and (d) show the corresponding topological gaps $\mathcal{Q}\Delta_{\rm gap}$, where $\mathcal{Q}$ is the scattering invariant and $\Delta_{\rm gap}$ is an estimator for the gap given by the energy of the second level. While the clean wire shows localized Majorana zero modes with a large topological gap ($Q=-1$, $\Delta_{\rm gap}>1\,E_{\rm so}$), both the localized zero modes and the topological phase are destroyed by strong disorder.
• Figure 3: Elements of the conductance matrix between lead $L$ and $R$ as a function of the Zeeman field $E_z$ and bias voltage $V_{\rm bias}$ at a chemical potential $\mu=1\,E_{\rm so}$ in a clean wire of length $L=32.5\,l_{\rm so}$. For evaluating the metric, a measurement of $G_{RR}$ and $G_{LL}$ at zero bias is performed (black square), and a scan of the non-local conductances along $E_z$ for zero bias and along $V_{\rm bias}$ for a given $E_z$ are needed (black lines). Increasing the Zeeman field $E_z$, the wire enters the topological phase which shows zero bias peaks in the local conductance at both leads. At the transition, the gap closes which can be inferred from the non-local conductance.
• Figure 4: Majorana wire with strong disorder (cf. Fig. \ref{['Fig:ref']}b) using optimized gate voltages found by minimizing the metric Eq. \ref{['Eq:metric']} with the CMA-ES algorithm. (a) Wave function $|\Psi_0|^2$ (blue) of the first level together with the electron $|\bm{u}_0|^2$ (green) and hole wave function $|\bm{v}_0|^2$ (orange), (b) topological gap $\mathcal{Q}\Delta_{\rm gap}$ , and (c) the energy levels for the optimized wire (blue) and for comparison for the disordered wire with zero voltage on all gates (red). Using optimized gate voltages restores localized MZMs, the topological phase, and a topological gap.
• Figure 5: Elements of the conductance matrix between lead $L$ and $R$ as a function of the Zeeman field $E_z$ and chemical potential $\mu$ for the disordered wire with optimized gate voltages (see Fig. \ref{['Fig:opt']}). The local conductances are shown at zero bias and the non-local conductances for a small bias voltage of $V_{\rm bias}=0.1\,E_{\rm so}$. Based on these conductance measurements, the topological gap protocol would indicate an extended topological phase with gap closing along the boundary. The black square indicates the point in the phase diagram at which the optimization has been performed.