How To Use Neural Networks To Investigate Quantum Many-Body Physics

Over the past few years, machine learning has emerged as a powerful computational tool to tackle complex problems in a broad range of scientific disciplines. In particular, artificial neural networks have been successfully used to mitigate the exponential complexity often encountered in quantum many-body physics, the study of properties of quantum systems built from a large number of interacting particles. In this article, we review some applications of neural networks in condensed matter physics and quantum information, with particular emphasis on hands-on tutorials serving as a quick start for a newcomer to the field. The prerequisites of this tutorial are basic probability theory and calculus, linear algebra, basic notions of neural networks, statistical physics, and quantum mechanics. The reader is introduced to supervised machine learning with convolutional neural networks to learn a phase transition, unsupervised learning with restricted Boltzmann machines to perform quantum tomography, and the variational Monte Carlo method with recurrent neural networks for approximating the ground state of a many-body Hamiltonian. For each algorithm, we briefly review the key ingredients and their corresponding neural-network implementation, and show numerical experiments for a system of interacting Rydberg atoms in two dimensions.

Popular Summary

Machine learning has recently emerged as a powerful computational paradigm to study complex problems in quantum many-body physics. This activity indicates that this method, specifically techniques based on neural networks, may soon become commonplace in quantum physics research, both in experiments and in numerical simulations. To stimulate researchers to familiarize themselves with the wealth of machine learning concepts, algorithms, and research culture, we have developed a set of basic hands-on tutorials focused on examples of applications of neural networks to problems in condensed matter physics and quantum computing.

Because of their high programmability and the possibility to access physical information through a large volume of measurement data, our applications focus on the analysis and numerical simulations of arrays of Rydberg atoms. These systems lend themselves exceptionally well to study with neural-network methods. In particular, we consider supervised learning of a phase transition, the reconstruction of a quantum state from measurements, and variational Monte Carlo simulations. For each algorithm, we briefly review the key ingredients and their corresponding neural-network implementation.

As machine learning algorithms continue to be adopted and repurposed in the research landscape of strongly correlated quantum matter and quantum information science, we hope this tutorial will provide a useful step into the expanding domain of the application of artificial intelligence to quantum many-body systems.

Figure 1

Rydberg atoms in a two-dimensional square array. (a) The phase diagram at a fixed value of the interaction $V=3\mathrm{MHz}$ and $\mathrm{\Omega }=1\mathrm{MHz}$. At large and negative detuning, the system is in a disordered phase with all atoms in the ground state. At large and positive detuning, the atoms are found in a checkerboard pattern with Néel order. (b) Snakelike geometry of the MPS path along the square lattice, used for the DMRG simulations. Ground-state energy (c) and the staggered magnetization (Néel order) (d) as a function of the detuning $\delta$ for an $8\times 8$ array ($V=3\mathrm{MHz}$). (e) Absolute value of the average occupation number in momentum space $|n(\mathit{k})|$ deep into the $Z_2$ ordered phase ($\delta =4\mathrm{MHz}$), showing a peak at $\mathit{k}=(\pi ,\pi )$, a signature of antiferromagnetic order. (f) Energy gap $\mathrm{\Delta }$ between the ground state and the first excited state, detecting a quantum phase transition at detuning $\delta \approx 1.3$. PM, paramagnetic.

Figure 2

A convolutional neural network. The elements of the input ${\mathsf{h}}_{l,j,k}^{(0)}$ correspond to the outcome of a projective measurement on the Rydberg system. The first operation is a convolutional layer with $M_y\times M_x=3\times 3$ kernels with $I_{\mathrm{out}}=32$ output channels and $L_{\mathrm{input}}=1$. This kernel is convolved with an input configuration with $N=8\times 8$ Rydberg atoms. Likewise, the second operation corresponds to a convolutional layer with $M_y\times M_x=3\times 3$ kernels with $I_{\mathrm{out}}=32$ output channels and $L_{\mathrm{input}}=32$. The output of the second convolutional layer is flattened and fed to a FC layer with a ReLU activation, followed by another FC layer with a softmax activation, which produces the prediction outcome.

Figure 3

Learning the quantum phase transition in the Rydberg-atom array. Top: Output signal for the two units in the topmost dense layer of the neural-network architecture as a function of the detuning, corresponding to Eq. (5). Bottom: Accuracy on the test dataset, showing a dip near the critical point. This estimator is calculated by comparing the prediction of the CNN with the correct labels assigned to the samples in the test set.

Figure 4

Quantum state tomography of an $8\times 8$ array of Rydberg atoms with unsupervised learning of single-shot atomic occupation data. We compare various observables measured using the MPSs obtained from DMRG (black line) and the neural-network wave functions learned from data generated at different values of the detuning $\delta$. We plot the average energy per spin (a) and the average magnetization along the $z$ axis (b) and $x$ axis (c). The inset in (a) shows the relative error in the energy ${ϵ}_{\mathrm{rel}}=|E_{\mathrm{RBM}}-E_{\mathrm{DMRG}}|/|E_{\mathrm{DMRG}}$. Error bars estimated via standard deviations are too small to be visible.

Figure 5

Quantum state tomography of the beryllium hydride molecule. We show the fidelity $\mathcal{F}=|⟨\mathrm{\Phi }|{\psi }_{\mathit{\theta }}⟩{|}^2$ between the neural-network wave function ${\psi }_{\mathit{\theta }}$ and the exact molecular wave function $\mathrm{\Phi }$ at each iteration during training.

Figure 6

Recurrent neural network. (a). A generic RNN cell (left) and its unrolling in time to process sequenced data (right). (b) The gated recurrent unit, and the set of operations used to process the input latent state ${\mathit{h}}_{j-1}$ and visible state $x_j$ to generate a new latent state ${\mathit{h}}_j$. Lines joining together means concatenation, while the circles are element-wise operations.

Figure 7

Variational Monte Carlo simulation of the Rydberg Hamiltonian. (a) Average energy as a function of the training step near the phase transition at $\delta =1.3$. We show RNN wave functions with $n_h=25$ and $n_h=100$ hidden units, compared with the result obtained from DMRG. The first few VMC iterations are not plotted to increase readability of the convergence transient. (b) Average energy over the full phase diagram for a RNN wave function with $n_h=100$, in comparison with DMRG.