Deep learning-enhanced variational Monte Carlo method for quantum many-body physics

Artificial neural networks have been successfully incorporated into the variational Monte Carlo method (VMC) to study quantum many-body systems. However, there have been few systematic studies exploring quantum many-body physics using deep neural networks (DNNs), despite the tremendous success enjoyed by DNNs in many other areas in recent years. One main challenge of implementing DNNs in VMC is the inefficiency of optimizing such networks with a large number of parameters. We introduce an importance sampling gradient optimization (ISGO) algorithm, which significantly improves the computational speed of training DNNs by VMC. We design an efficient convolutional DNN architecture to compute the ground state of a one-dimensional SU($N$) spin chain. Our numerical results of the ground-state energies with up to 16 layers of DNNs show excellent agreement with the Bethe ansatz exact solution. Furthermore, we also calculate the loop correlation function using the wave function obtained. Our work demonstrates the feasibility and advantages of applying DNNs to numerical quantum many-body calculations.

Figure 1

The architecture of a convolutional DNN with $L=8$ hidden layers. The input state is encoded into a 2D tensor of shape $[N_{\text{site}},N_{\text{in}}]$, and fed into the input layer (represented by the leftmost pink rectangle). For value encoding $N_{\text{in}}=1$ and for one-hot encoding $N_{\text{in}}=N$. The blue rectangles stand for the activation (feature maps) of the hidden layers. Convolution filters (the small pink rectangles) transform one hidden layer to the next one. The last hidden layer (on the right) is reduce summed and followed by a fully connected layer to give the $ln\mathrm{\Psi }$ output.

Figure 2

The flowchart comparing the conventional VMC algorithm and the VMC with ISGO algorithm.

Figure 3

(a) The flowchart of the ISGO algorithm within the VMC method. A network is used to represent the ground-state wave function. In every iteration step, the sampler generates $N_{\text{sample}}$ samples following distribution $P_x^0\propto {\mathrm{\Psi }}_x^{*0}{\mathrm{\Psi }}_x^0$. Then the importance sampling optimizer updates the network parameters through back propagation in a loop for $N_{\text{optimize}}$ times. $N_{\text{optimize}}=1$ corresponds to the conventional gradient optimization method. The whole process is iterated until convergence. The sampler and the optimizer share the same wave function. We compare the training curves for a $N_{\mathrm{site}}=60$ SU(2) spin chain using a $(L,F,K)=(16,8,3)$ CNN with one-hot encoding. (b) Variational energy vs iteration steps. (c) Variational energy vs wall time on GPU/CPU. The initial learning rate for Adam is ${10}^{-4}$.

Figure 4

The ground-state energy for $N_{\text{site}}=60$ SU($N$) spin chains ($N=2,3,4,5$) using CNN with (a), (b) value encoding and (c), (d) one-hot encoding. (a) and (c) are for one layer $L=1$ with fixed channel number $F=8$ and different kernel size $K$. (b) and (d) are for fixed channel number and kernel size ($F=8$, $K=3$) but different number of layers $L$. In (b), the black squares are for $F=8$ and $K=5$. The horizontal black dashed lines are exact results from the Bethe ansatz.

Figure 5

(a) Real-space loop correlation functions $S_r$ for the SU($N$) spin chain with $N=5,4,3,2$ from top to bottom. (b) The Fourier transform of $S_r$: $S_k=\left|{\sum }_r{S}_r{e}^{ikr}\right|$ with peaks at $k=\pm \pi /N$.