Machine learning Majorana topology using unsupervised and supervised learning

Jacob Taylor; Haining Pan; Sankar Das Sarma

cond-mat.dis-nnDec 15, 2025

Machine learning Majorana topology using unsupervised and supervised learning

Authors: Jacob Taylor, Haining Pan, Sankar Das Sarma

Organizations: Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA · Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, NJ 08854, USA

Abstract

In unsupervised learning, the training data for deep learning does not come with any labels, thus forcing the algorithm to discover hidden patterns in the data for discerning useful information. This, in principle, could be a powerful tool in identifying topological order since topology does not always manifest in obvious physical ways (e.g., topological superconductivity) for its decisive confirmation. The problem, however, is that unsupervised learning is a difficult challenge, necessitating huge computing resources, which may not always work. In the current work, we combine unsupervised and supervised learning to establish that unlabeled (simulated) data in the Majorana splitting in realistic short disordered nanowires may enable not only a distinction between topological' and trivial', but also where their crossover happens in the relevant parameter space. This may be a useful tool in identifying topology in experimental Majorana nanowires.

Figures & tables

effective mass $m^{*}$	$0.01519~m_{e}$
chemical potential $\mu$	$1~\mathrm{meV}$
spin-orbit coupling $\alpha$	$0.5~\mathrm{eV\AA}$
constant SC gap $\Delta$	$0.2~\mathrm{meV}$
wire length $L$	$0.6$ – $15~\mu\mathrm{m}$
Gaussian disorder strength $\sigma$	$0.1\text{--}6~\mathrm{meV}$
temperature	$0$

Table 1: Parameters in the Majorana nanowire simulations adapted from [ 40 ] .

Panel of Figure 1 — Figure 1: Autoencoder architecture used for dimensionality reduction in the unsupervised learning framework. The network consists of a 1D convolutional encoder (with stride size of 2) and decoder with a symmetric architecture in reverse order. Left column shows examples of input data: Majorana energy splittings $E_{s}(V_{z})$ and topological visibility $\text{TV}(V_{z})$ . The encoder compresses these into a 15-dimensional latent vector, which the decoder reconstructs back to the original input format. Training minimizes the mean-squared error between input and reconstructed output.

Figure 2: Top row: Latent space visualization from the auto-encoder principal component analysis (PCA), displaying the first three principal components (PC1, PC2, and PC3) for (a) $k=2$ and (b) $k=3$ cluster classifications obtained from $k$ -means clustering. Each data point represents one disorder realization, compressed from the complete $E_{s}(V_{z})$ and $\text{TV}(V_{z})$ curves via the autoencoder in Fig. 1 . Bottom row: Physical parameters associated with each latent space point: (c) disorder strength $\sigma$ and (d) wire length $L$ .

Figure 3: Top row: Unsupervised phase diagrams in the disorder-length (

\sigma

L

) parameter space, obtained by mapping latent space clusters back to physical parameters. (a) Two-cluster classification reveals a phase boundary between weak-disorder (Cluster 1, green) and strong-disorder (Cluster 0, blue) regimes. (b) Three-cluster classification identifies an additional intermediate regime (Cluster 2, orange) at the crossover. Each point is labeled by the majority cluster among 100 disorder realizations. (c) Silhouette score as a function of cluster number

k

, showing that

k=2

and

k=3

have comparable quality, but the score drops sharply beyond

k=3

. Bottom row (d-f): Same clustering analysis for restricted disorder range

\sigma\in[0,1]

meV, where (f) indicates optimal clustering at

k=2

Figure 4: Unsupervised phase diagrams in disorder-length ( $\sigma$ - $L$ ) space, obtained by mapping latent-space clusters back to physical parameters using $E_{s}$ and $L$ as inputs (but only rewarding $E_{s}$ reconstruction). (a) Four-cluster classification separates weak-disorder, strong-disorder, intermediate-disorder, and short-length regimes; each point is labeled by the majority cluster over 100 disorder realizations. (b) Silhouette score versus cluster number $k$ , showing optimal clustering at $k=4$ with a drop for larger $k$ .

Panel of Figure 5 — Figure 5: (a) Architecture of the supervised learning neural network based on 1D convolutional encoder-decoder. Inputs are experimentally measurable quantities ( $E_{s}(V_{z})$ and wire length $L$ ); outputs are theoretically accessible quantities (topological visibility $\text{TV}(V_{z})$ and disorder strength $\sigma$ ) that are not directly measurable in experiments. (b) Predicted versus ground-truth topological visibility for the full disorder range $\sigma\in[0,6]$ meV, showing good correlation along the diagonal (red dashed line indicates perfect prediction). (c) Predicted versus ground-truth disorder strength for $\sigma\in[0,6]$ meV; accuracy is higher at weak disorder and decreases at strong disorder. (d, e) Same predictions for the restricted weak-disorder range $\sigma\in[0,1]$ meV, demonstrating significantly improved accuracy for both TV and $\sigma$ .

Supplementary Figure A1: Explanatory diagram of Supervised learning Neural Network. The neural network consists of 5 parts: the $E_{s}(V_{z})$ encoder (in green) that converts the splitting into a smaller embedding space, the global context vector (GCV) (in purple) that converts the embedding along with the length into a small context vector containing global information able to be used as a conditional, the local modification from GCV (in black) that takes the GCV and modifies the embedding space in the locally relevant manner, the decoder (in red) which expands and maps the embedding along with the GCV conditional into the $\text{TV}(V_{z})$ output, and finally the small dense neural network in (orange) that converts the GCV to a global disorder magnitude.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Supplementary Figure A2: Unsupervised phase diagram for input data lacking TV in the disorder-length ( $\sigma$ - $L$ ) parameter space, obtained by mapping latent space clusters back to physical parameters. Two-cluster classification reveals trivial clustering only between short and long systems, with some noise yielding to unclean boundaries. Forcing more than 2 clusters yields unpredictable results.

Supplementary Figure A3: Unsupervised phase diagrams in disorder-length ( $\sigma$ - $L$ ) space, obtained by mapping latent-space clusters back to physical parameters using $E_{s}$ and $L$ as inputs (but only rewarding $E_{s}$ reconstruction). (a) Three-cluster classification separates weak-disorder, strong-disorder, and short-length regimes; each point is labeled by the majority cluster over 100 disorder realizations. See Fig. 4 for four-cluster results and silhouette scores.

Supplementary Figure A4: Prediction error heatmaps in the ( $\sigma$ , $L$ ) parameter space for the full disorder regime $\sigma\in[0,6]$ meV: (a) TV prediction error, (b) disorder magnitude prediction error. Color indicates the magnitude of prediction error (lighter/yellow = higher error). Interestingly, errors appear larger near the phase transition resulting from the unsupervised learning.

Supplementary Figure A5: Prediction error heatmaps in the ( $\sigma$ , $L$ ) parameter space for weak disorder regime $\sigma\in[0,1]$ meV: (a) TV prediction error, (b) disorder magnitude prediction error. Color indicates the magnitude of prediction error (lighter/yellow = higher error). Errors are generally larger in the high disorder, short wire length regime.

Supplementary Figure A6: Semi-supervised phase diagrams in the disorder-length ( $\sigma$ - $L$ ) parameter space, obtained by mapping latent space clusters back to physical parameters. The input consisted entirely of $E_{s}$ , with TV being predicted by the supervised network. Both $E_{s}$ and the predicted TV were then fed into the unsupervised process. Results are in good qualitative agreement with those obtained using the exact TV values. (a) Two-cluster classification reveals a phase boundary between weak-disorder (Cluster 1, green) and strong-disorder (Cluster 0, blue) regimes. (b) Three-cluster classification identifies an additional intermediate regime (Cluster 2, orange) at the crossover. Each point is labeled by the majority cluster among disorder realizations.

Supplementary Figure A7: Average $R^{2}(\text{TV})$ vs. $|\delta|$ , where $|\delta|$ is the standard deviation of additive error $E_{s}^{\prime}(V_{z})=E_{s}(V_{z})+\delta(V_{z})$ for (a) the standard neural network (b) the neural network with error added during training. The neural network shows robust resistance to measurement errors within $E_{s}(V_{z})$ up to (a) $5\mu eV$ and (b) $50\mu eV$ .

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