Revealing quantum chaos with machine learning

Understanding properties of quantum matter is an outstanding challenge in science. In this paper, we demonstrate how machine-learning methods can be successfully applied for the classification of various regimes in single-particle and many-body systems. We realize neural network algorithms that perform a classification between regular and chaotic behavior in quantum billiard models with remarkably high accuracy. We use the variational autoencoder for autosupervised classification of regular/chaotic wave functions, as well as demonstrating that autoencoders could be used as a tool for detection of anomalous quantum states, such as quantum scars. By taking this method further, we show that machine-learning techniques allow us to pin down the transition from integrability to many-body quantum chaos in Heisenberg XXZ spin chains. For both cases, we confirm the existence of universal W shapes that characterize the transition. Our results pave the way for exploring the power of machine-learning tools for revealing exotic phenomena in quantum many-body systems.

Figure 1

Neural network approach for identifying a transition between chaotic and regular states in quantum billiards and Heisenberg spin chains. The input data contains probability distribution in the configuration space, the two neuron activation functions are used for the identification of the two regimes.

Figure 2

Convolutional neural network outputs for (a) Sinai billiard, (b) Bunimovich stadium, and (c) Pascal's limaçon as functions of the chaoticity parameter $\lambda$ characterizing the billiard's boundary shape. The highlighted critical region corresponds to the regions of “uncertainty” in neuron network output activation curves.

Figure 3

Autosupervised learning of regular and chaotic states in quantum billiards with variational autoencoder (VAE). Latent space representation of the wave functions in (a) Bunimovich stadium, (b) Sinai billiard; $z_{1,2}$ are coordinates in the two-dimensional latent space. (c) Anomaly detection: Latent space representation of wave functions corresponding to regular (red dots, $\lambda =0$) and chaotic (blue dots, $\lambda =0.4$) wave functions as well as scarred chaotic wave functions (green dots) in Bunimovich billiard. (d) VAE as a generative model: Images of wave functions ${\left|\psi \right|}^2$ generated by VAE corresponding to different positions in the latent space variables $(z_1,z_2)$ (Pascal billiard). By continuously scanning across two-dimensional latent space, the VAE performs a smooth interpolation between wave functions from chaotic and regular wave functions.

Figure 4

Neural network classification accuracy between integrable and chaotic XXZ spin chains with the next-nearest-neighbor interactions for $N=15$ spins ($J_{zz}=1$). (b) Distribution of energy-level spacings and the Poisson/Wigner-Dyson distributions. Plots correspond to XXZ model. (c) Learning by confusion W-like NN performance curve.

Figure 5

Left panel: NN classification accuracy for chaos-integrability transition in XXZ model in the presence of a local magnetic field (a magnetic impurity) for $N=15$ spins. Right panel: Energy-level spacing distributions for different values of $\lambda$.

Figure 6

Sample images of ${\left|\psi (x,y)\right|}^2$ in the data set for Bunimovich billiard. Regular case ($\lambda =0$) and chaotic case ($\lambda =l/r=0.2$).

Figure 7

CNN used for recognizing chaotic regimes in quantum billiards.

Figure 8

Left column: The CNN activation functions (Fig. 1 from the main text). The histograms show the energy-level spacing distributions (lowest 500 energy levels). To compare NN predictions for the regular-to-chaos critical region, we compare the energy-level spacing distribution with the standard Poisson/GOE distributions.

Figure 9

Architecture of variational autoencoder (VAE) for unsupervised learning of regular-chaos transition in quantum billiards.

Figure 10

Left panel: Examples of scarred wave functions in Bunimovich and Sinai billiards used for quantum anomaly detection. Right panel: Latent space distribution of VAE for regular ($\lambda =0$) and chaotic ($\lambda =0.4$) wave functions in Sinai billiard. Green dots correspond to scared wave functions.

Figure 11

Multilayer perceptron used for investigation integrable/chaotic transitions in Heisenberg XXZ chains.

Figure 12

Universal W-like NN performance curves in the learning by confusion scheme for the Sinai billiard (a), the Pascal's limaçon (b), XXZ $+$ NNN (c), and XXZ $+$ Impurity (d). The predicted transition point ${\lambda }_c$ is highlighted. The estimated position of the transition point predicted from the KL divergence calculation is shown with a red dot.