Machine learning discovery of new phases in programmable quantum simulator snapshots

TL;DR

This work addresses the challenge of extracting detailed phase information from large datasets generated by programmable quantum simulators. It introduces Hybrid-CCNN, a two-stage framework that first uses unsupervised Fourier-space phase discovery to seed a rough phase diagram and then applies interpretable CCNNs to density-fluctuation maps to sharpen phase boundaries and identify phase identities. The approach discovers five phases—two previously undetected (the rhombic and boundary-ordered phases)—and provides interpretable correlator motifs that align with known order parameters while revealing quantum-fluctuation features in the striated phase and finite-size signatures of the rhombic phase. By training entirely on experimental data, the method demonstrates a powerful data-centric path to uncover and characterize complex quantum states, with potential to reveal entanglement structures and guide exploration of exotic matter in PQS platforms.

Abstract

Machine learning has recently emerged as a promising approach for studying complex phenomena characterized by rich datasets. In particular, data-centric approaches lend to the possibility of automatically discovering structures in experimental datasets that manual inspection may miss. Here, we introduce an interpretable unsupervised-supervised hybrid machine learning approach, the hybrid-correlation convolutional neural network (Hybrid-CCNN), and apply it to experimental data generated using a programmable quantum simulator based on Rydberg atom arrays. Specifically, we apply Hybrid-CCNN to analyze new quantum phases on square lattices with programmable interactions. The initial unsupervised dimensionality reduction and clustering stage first reveals five distinct quantum phase regions. In a second supervised stage, we refine these phase boundaries and characterize each phase by training fully interpretable CCNNs and extracting the relevant correlations for each phase. The characteristic spatial weightings and snippets of correlations specifically recognized in each phase capture quantum fluctuations in the striated phase and identify two previously undetected phases, the rhombic and boundary-ordered phases. These observations demonstrate that a combination of programmable quantum simulators with machine learning can be used as a powerful tool for detailed exploration of correlated quantum states of matter.

Figures (15)

• Figure 1: (a) Defect-free square lattices of neutral atoms undergo coherent quantum evolution for different values of blockade extent $R_b/a$ and linear detuning sweeps' endpoints $\Delta/\Omega$, followed by projective readout in which atoms excited to the Rydberg state are detected as loss (red circles). (b--e) Idealized real-space patterns corresponding to phases predicted to be present at various regions of parameter space. Dark pink and white sites indicate $|r\rangle$ and $|g\rangle$ states, respectively, while the light pink sites in the striated phase are in a quantum superposition of $|r\rangle$ and $|g\rangle$. (f) A diagram outlining the Hybrid-CCNN approach. First, an unsupervised technique is used to generate a rough first-pass phase diagram. Here, we choose to measure average Fourier amplitudes $\overline{|n({\boldsymbol{k}})|^2}$ at each $(\Delta, R_b)$, perform a dimensionality reduction using principal component analysis, and finally cluster using a Gaussian mixture model. The resulting phase diagram informs the starting "seeds" in the parameter space, from which snapshots are sampled in a second supervised stage. We then learn to distinguish these snapshots using interpretable classifiers, from which we can extract refined phase boundaries and key identifying features.
• Figure 2: (a--c) Principal components $1, 2, 5$ in Fourier space. (d) Results of the GMM clustering performed in the reduced $10$-dimensional PCA space and projected for visualization into the space spanned by (a--c). (e) The CCNN architecture for the supervised learning stage, constructed here up to third order, $C_\alpha^{(3)}$, with three learned filters $f_\alpha$ and a learned spatial weighting $w({\boldsymbol{x}})$. First, a density-normalized snapshot $\delta n({\boldsymbol{x}})$ is convolved with the filters $f_\alpha$ to produce a convolutional map $C_\alpha^{(1)}({\boldsymbol{x}})$. The $\delta n({\boldsymbol{x}})$ are zero-padded to allow a convolution of the filters over the entire snapshot. Then, a series of polynomials are applied [Eqs. \ref{['eq:ccnn-C2']},\ref{['eq:ccnn-C3']}] to produce maps $C_\alpha^{(m)}({\boldsymbol{x}})$, which measure $m$th-order correlators near each ${\boldsymbol{x}}$. These maps are summed with a learned spatial weighting $w({\boldsymbol{x}})$ to produce features $c_\alpha^{(m)}$, which are used by a final logistic layer for classification. (f) Resulting phase diagram produced by supervised learning, obtained by cropping the classification confidence maps at level-set contours $\hat{y}=0.75$ and overlaying them.
• Figure 3: (a) CCNN-learned region of support for the striated phase, with highlighted boxes indicating the training points. (b) Previously used approximate order parameter detecting the striated phase. Red markers indicate phase boundaries obtained from DMRG simulations on a $9\times 9$ array Ebadi2021Nature. (c) The filters learned by a third-order nonuniform CCNN to identify the striated phase in (a) and the signs on the $\beta_\alpha^{(2)}$ coefficients connecting the corresponding $c_\alpha^{(2)}$ to the output. For ease of display, the filter weights are normalized such that the maximum is $1$ within each filter. (d) The spatial weighting $w({\boldsymbol{x}})$ learned by a third-order CCNN identifying the striated phase. A single-pixel outer layer, corresponding to where the filter lands on the zero-padded region, is omitted for clarity. (e) A diagram showing example patches of the idealized striated phase whose correlations are measured by the CCNN of (c,d).
• Figure 4: (a) CCNN-learned region of support for the edge phase, with highlighted boxes indicating the training points. (b) The learned spatial weighting and filters for the model trained to identify the edge-ordered phase, with the spatial extent of the snapshot indicated by the dashed line. The outermost pixels correspond to where the filter is centered on zero-padding but "clip" the edge pixels of the snapshot. For display purposes, the filter weights are normalized such that the maximum is $1$ within each filter. (c) Measured performance discrepancy between second-order models with a fixed uniform $w({\boldsymbol{x}})$$=$$1$ and a freely learned spatial weighting. The central lines and bands show the mean and standard deviation across five randomly initialized models of each type, respectively. (d) Experimental snapshots from the training set, showing $\bullet\circ\bullet$ motifs primarily only along the single-site border, with the interior highly disordered. (e) Measurement of the third-nearest-neighbor $\langle \delta n_{i,j} \delta n_{i+2, j}\rangle$ connected correlation function within the edge and bulk (all sites but the outermost two-site strips) along a cut at $R_b$$=$$1.46$, averaged across translations and other symmetries.
• Figure 5: (a) The region of support for the rhombic phase as learned by the full CCNN model of Fig. 1(n), with $5\times 5$ filters and highlighted boxes indicating the training points. (b,c) The two learned $4$$\times$$4$ convolutional filters for a simplified model ($w({\boldsymbol{x}}) = 1, \beta_\alpha^{(n)} \ge 0$) trained to identify the rhombic phase. (d--i) High-weight two- and three-point connected correlators measured by $c_\alpha^{(2)}, c_\alpha^{(3)}$ resulting from the filters in (a,b). We find $\beta_2^{(2)}$ to be nearly zero, so we omit two-point correlators stemming from the filter $f_2$. Our CCNN is symmetrized, (see Appendix B) and so measures all correlators symmetry-equivalent under rotations and flips to those shown. (j) Identification of these two- and three-point motifs in the idealized rhombic ordering with boundary defects (light blue). (k) Identification of local occurrences of these motifs in experimental snapshots sampled from the training set.