Unsupervised generalization of correlated quantum dynamics on disordered lattices

The application of unsupervised machine learning to the study of physical problems is an emerging field aimed at leveraging their outstanding capabilities to distill essential information from complex data sets. Here we use Generative Adversarial Networks (GANs), deep learning algorithms mainly used to synthesize realistic images, for the analysis of a physical problem: the propagation of interacting quantum particles on disordered chains, characterized by the emergence of complex quantum correlations. We find that GANs can learn these correlations and identify the physical control parameters in a completely unsupervised manner. After training on a data set of unlabeled examples, the algorithm can generate, without further calculations, a much larger number of unseen yet physically correct new instances. Most notably, the knowledge distilled in the algorithm's latent space identifies the relevant control parameter, the level of disorder in our case, even though this was not highlighted in the training set. This allows tuning the disorder level in the generated samples to values the algorithm was not explicitly trained on, thereby uncovering a fundamental physical phenomenon—the crossover from nearly free propagation to localization. These results demonstrate the ability of deep neural networks to learn and generalize physical rules by scanning examples and suggest their use in physical analysis and discovery.

Figure 1

Correlated quantum walks and training samples. (a) Illustration of correlated, two-particle quantum walk on a disordered lattice, showing the evolution of the particle density. (b) An example of the resulting two-particle correlation function—the positions of the two particles at the end of the propagation depend on each other. [(c), (d)] Examples of the training samples. The left column of pixels represents the disordered potential, $E_n$. The matrix on the right represents the resulting two-particle correlation for specific initial state, or the full unitary transformation.

Figure 2

GANs can generate new yet physically correct examples. (a) Improvement in generating physically correct correlations as learning progresses, compared to exact calculation (last column) for (i) $\mathrm{\Gamma }=0$, no interactions; (ii) $\mathrm{\Gamma }=3$, intermediate interactions; and (iii) $\mathrm{\Gamma }=1000$, strong interactions. Results are for $|{\psi }_{\text{initial}}\rangle =a_5^{†}{a}_6^{†}|0\rangle$, averaged over 1000 realizations of disorder. (b) The physical validity of the generated correlation functions as a function of the learning length, quantified using the Kullback-Leibler divergence. Blue overlay represents standard deviation across disorder realizations.

Figure 3

Latent-space truncation trick tunes the level of disorder and reveals the localization crossover. (a) Histograms of generated on-site energies from 1000 samples as a function of the truncation parameter $\delta$. While the original network was trained on disorder level corresponding to the rightmost histogram, truncating the latent space produces different levels of disorder. (b) Disorder-averaged correlation functions (top) and densities (bottom) for two particles initially on the same site and zero interaction. (c) Disorder-averaged correlation functions (top) and densities (bottom) for two particles initially on adjacent site an $\mathrm{\Gamma }=3$. All results are physically valid with Kullback-Leibler divergence below 0.03.

Figure 4

Tuning full unitary transformations: latent-space truncation used to tune disorder level in the generated two-particle unitaries. (a) Fidelity of the generated samples during learning, for a GAN network starting from scratch (circles) and for a pretrained network (squares); see text. (b) (i) Histograms of generated on-site energies from 1000 samples as a function of the truncation parameter $\delta$. (ii) Disorder-averaged two-particle unitaries (magnitude).