Shuttling Majorana zero modes in disordered and noisy topological superconductors

Abstract

The braiding of Majorana zero modes (MZMs) forms the fundamental building block for topological quantum computation. Braiding protocols which involve the physical exchange of MZMs are typically envisioned on a network of topological superconducting wires. An essential component of these protocols is the transport of MZMs, which can be performed by using electric gates to locally tune sections of the wire between topologically trivial and non-trivial phases. In this work, we numerically simulate this transport by tuning a single section of a superconducting wire which contains either disorder (uncorrelated and correlated) or noise. We focus on the impact of these additional effects on the diabatic error, which describes unwanted transitions between the ground state and excited states. We show that the behavior of the average diabatic error is predominantly controlled by the statistics of the minimum bulk energy gap which is suppressed in the presence of disorder. The increase in diabatic error can be several orders of magnitude and is most deleterious when the disorder correlation length is a finite fraction of the transport distance and negligible when these lengths are far apart. In the presence of noise, the diabatic error is significantly enhanced due to optical transitions which depend on the minimum bulk energy gap as well as the frequency modes present in the noise. The results presented here serve to further characterize the diabatic error in disordered and noisy settings, which are important considerations in practical implementations of physical braiding schemes.

Figures (9)

• Figure 1: Illustration of the protocol used to transport a MZM across a topological superconducting wire over a distance $R$ in time $\tau$ by tuning local chemical potentials. The wire contains two MZMs, $\gamma_{\mathrm{A}}$ and $\gamma_{\mathrm{B}}$, with the latter undergoing transport toward the wire's right edge. The MZM wave functions are pictorially depicted at each major step of the protocol. (i) Initial configuration of the wire, which is divided into a left section that is topologically nontrivial and a right section which is trivial. (ii) Configuration of the wire when the right section reaches the critical point resulting in the delocalization of $\gamma_{\mathrm{B}}$ across the section. (iii) End of the transport protocol with the wire being completely in the nontrivial phase and with $\gamma_{\mathrm{B}}$ located on the wire's right edge.
• Figure 2: Numerical results for the diabatic error at the end of the transport protocol simulated in a chain containing uncorrelated disorder on the chemical potential. Results for the average error are displayed as solid markers with solid lines while those for the typical error are displayed as open markers with dashed lines. (a) Diabatic error versus protocol time for select disorder strengths. (b) Diabatic error versus disorder strength with protocol time $\tau/\tau_{\mathrm{LZ}} = 15$. (c) Diabatic error versus protocol time for select transport distances$R$ with disorder strength $\sigma/w = 0.05$. (d) Comparison between the numerical result for the diabatic error with disorder strength $\sigma/w = 0.05$ and the corresponding semianalytical expression which uses Eq. (\ref{['eq:diaberr_analytical']}), actual values of the minimum gap extracted from simulations, and averaging. Unless otherwise specified, the default chain parameters are $L = R = N/2 = 30$ sites, $\mu_{0} = 0.4$ meV, $w = 3$ meV, and $\Delta_{\mathrm{SC}} = 0.6$ meV. For each result, averaging is performed over $500$ simulations with different disorder realizations.
• Figure 3: Numerical results for the minimum gap statistics of the transport protocol simulated in a chain with uncorrelated disorder on the chemical potential. (a) Average minimum gap versus disorder strength for select transport distances$R$. The clean minimum gap $\Delta_{R}$ is subtracted from each result to demonstrate the absence of any $R$-dependent contributions from the disorder. (b) Standard deviation of the minimum gap versus disorder strength for select $R$. The results are scaled with respect to $\sqrt{R}$ to demonstrate the $\sim 1/\sqrt{R}$ dependence for weak disorder. (c) Normalized probability densities corresponding to select disorder strengths. The clean minimum gap value $\Delta_{R}$ is highlighted by a vertical dashed black line. Unless otherwise specified, the default chain parameters are identical to those cited in Fig. \ref{['fig:uncorr_diab']} and in the main text. For panels (a) and (b), each result is obtained by averaging over $1500$ simulations with different disorder realizations. For panel (c), each density is constructed by sampling $1500$ values of the minimum gap.
• Figure 4: Numerical results for the diabatic error at the end of the transport protocol simulated in a chain containing Gaussian-correlated disorder on the chemical potential. Results for the average error are displayed as solid markers with solid lines while those for the typical error are displayed as open markers with dashed lines. (a) Diabatic error versus protocol time for select correlation lengths with disorder strength $\sigma/w = 0.025$. (b) Diabatic error versus correlation length for select disorder strengths with protocol time $\tau/\tau_{\mathrm{LZ}} = 15$. (c) Diabatic error versus protocol time for select transport distances$R$ with disorder strength $\sigma/w = 0.025$ and correlation length $\xi/R = 0.2$. (d) Numerical result for the diabatic error with disorder strength $\sigma/w = 0.025$ and correlation length $\xi/R = 0.2$ in comparison to the corresponding semianalytical expression. Unless otherwise specified, the default chain parameters are identical to those cited in Fig. \ref{['fig:uncorr_diab']} and in the main text. Averaging is performed over $500$ simulations for each result.
• Figure 5: Numerical results for the minimum gap statistics of the transport protocol simulated in a chain with Gaussian-correlated disorder on the chemical potential. Results corresponding to select transport distances $R$ are displayed in each plot. (a) Average and (b) standard deviation of minimum gap versus disorder strength with correlation length $\xi/R = 0.2$. (c) Average and (d) standard deviation of minimum gap versus correlation length with disorder strength $\sigma/w = 0.025$. The horizontal dashed black line in panel (c) indicates the clean minimum gap value $\Delta_{R}$. To highlight certain scaling behaviors, $\Delta_{R}$ is subtracted off of the averages while the standard deviations are multiplied by $\sqrt{R}$. Unless otherwise specified, the default chain parameters are identical to those cited in Fig. \ref{['fig:uncorr_diab']} and in the main text. Averaging is performed over $1500$ simulations for each result.