Machine-learned tuning to protected states by probing noise resilience

TL;DR

This work presents a machine-learning method for tuning to protected regimes, based on injecting noise into the system and searching directly for the most noise-resilient configuration, including but not limited to isolated Majorana bound states.

Abstract

Protected states are promising for quantum technologies due to their intrinsic resilience against noise. However, such states often emerge at discrete points or small regions in parameter space and are thus difficult to find in experiments. In this work, we present a machine-learning method for tuning to protected regimes, based on injecting noise into the system and searching directly for the most noise-resilient configuration. We illustrate this method by considering short quantum dot-based Kitaev chains which we subject to random parameter fluctuations. Using the covariance matrix adaptation evolutionary strategy we minimize the typical resulting ground state splitting, which makes the system converge to a protected configuration with well-separated Majorana bound states. We verify the robustness of our method by considering finite Zeeman fields, electron-electron repulsion, asymmetric couplings, and varying the length of the Kitaev chain. Our work provides a reliable method for tuning to protected states, including but not limited to isolated Majorana bound states.

Figures (4)

• Figure 1: Tuning to MBS sweet spots via parameter fluctuations. (a) Representation of a QD-based Kitaev chain with an arbitrary number of sites. We inject local noise ($\eta_i$) into the QDs as an optimization algorithm searches for the point in parameter space most stable against them. (b) Pictorial representation of the energy splitting $E_0$ as a function of the detunings represented by the set $\{\boldsymbol{\eta}\}$ for different points in parameter space. For a MBS sweet spot (in blue), $E_0$ remains robust (to a certain degree) upon variations in the QD levels, while for non-sweet spots (in orange), the degeneracy of the ground state splits.
• Figure 2: Results for the 2-site Kitaev chain. (a,b) Energy splitting $E_0$ and MP at the outer QDs, as a function of the normal and superconducting QD levels, $\varepsilon_{1, 3}$ and $\varepsilon_{2}$, respectively. The blue crosses indicate sweet spots found by the automated tuning procedure. (c,d) Median values (lines) and standard deviations (shades) of (c) the three QD potentials and (d) MBS properties, i.e., MPs $M_1$ (blue) and $M_3$ (green), $E_{\rm ex}$ (purple), and $E_0$ (red), corresponding to the best members of each generation across $50$ independent simulations.
• Figure 3: Results for the 3-site Kitaev chain. (a) Sketch of the system. (b--d) Median values (lines) and standard deviations (shades) of the MBS properties [(b) MP $|M_1| = |M_5| = |M|$, (c) $E_{\rm ex}$, and (d) $E_0$] corresponding to the best members of each generation across $50$ independent simulations. The simulations were performed with $\beta = 1$ (green) and $\beta=2$ (purple). All parameters are the same as in Fig. \ref{['Fig2']}, apart from $W = 0.1\Delta$.
• Figure 4: Results for longer Kitaev chains. Evolution of the median values (lines) and standard deviations (shades) of $|M_{1}| = |M_N| = |M|$ (blue), $E_0$ (red), and $E_{\rm ex}$ (purple) corresponding to the best members of each generation across $50$ independent simulations, for (a) 4- and (b) 5-site Kitaev chains. All parameters are the same as in Fig. \ref{['Fig3']}, apart from $W= 0.125\Delta$.