Approximating quantum many-body wave functions using artificial neural networks

In this paper, we demonstrate the expressibility of artificial neural networks (ANNs) in quantum many-body physics by showing that a feed-forward neural network with a small number of hidden layers can be trained to approximate with high precision the ground states of some notable quantum many-body systems. We consider the one-dimensional free bosons and fermions, spinless fermions on a square lattice away from half-filling, as well as frustrated quantum magnetism with a rapidly oscillating ground-state characteristic function. In the latter case, an ANN with a standard architecture fails, while that with a slightly modified one successfully learns the frustration-induced complex sign rule in the ground state and approximates the ground states with high precisions. As an example of practical use of our method, we also perform the variational method to explore the ground state of an antiferromagnetic $J_1\text{−}{J}_2$ Heisenberg model.

Figure 1

(Left) The structure of the feed-forward ANN we used to approximate quantum many-body ground-state wave functions; (right) three types of neurons with different activation functions.

Figure 2

(a) Fidelity $F=1-|\langle {\mathrm{\Psi }}_{ED}|{\mathrm{\Psi }}_{\text{ANN}}\rangle |$ as a function of the number of neurons $N_b$, for 1D free bosons with $L=12$, $N=12$, and the corresponding Hilbert space dimensionality $\mathbb{D}=1352078$ and for 1D free fermions with $L=24$, $N=11$, and $\mathbb{D}=2496144$. (b) The precision function $\delta =|O_{\text{ANN}}-O_{ED}|/O_{ED}$ for two different quantities $O_{nn}=\frac{1}{L}{\sum }_i\langle n_i{n}_{i+1}\rangle$ and $O_{n1}=\langle n_1\rangle$ for 2D free fermions with $L=24$($4\times 6$), $N=13$, and $\mathbb{D}=2496144$, the inset shows a typical Fock configuration. (c) The precision function $\delta$ for $O_{nn}=\frac{1}{L}{\sum }_i\langle n_i{n}_{i+1}\rangle$ as a function of $N_b$ obtained by the importance sampling algorithm for a 1D free fermion, where both the amount of the training and the sampling data are chosen as $\mathbb{H}={10}^6$ in the simulations of various system sizes; the inset shows $\delta$ as a function of $L$ with a fixed $\mathbb{H}={10}^6$ and $N_b=200$.

Figure 3

(a) The accuracy function $P\left(\alpha \right)$ of the sign rule in the ground state of 1D $J_1\text{−}{J}_2$ model with $L=24$ and various $N_b$, with the inset showing the modified ANN used to capture the sign rule with neurons in the first hidden layer replaced by those with the cosine activation functions $f\left(x\right)=cos\left(\pi x\right)$. (b) The evolution of $P$ during the training with fixed $N_b=200$ and various systems sizes, with the inset showing a typical “confusion” time $T_c$ in the training as a function of system size with $N_b=200$. (c) The accuracy $P$ predicted by the sign ANN as a function of the layer of the ANN with a fixed $N_b=120$. (d) The ratio between the average absolute value of the coefficient with the erroneously assigned sign and that of the whole training set with $N_b=120$. (e) The fidelity as a function of $N_b$ for the amplitude ANN. (f) The fidelity in the $J_1\text{−}{J}_2$ model and various $\alpha$ as a function of $N_b$ predicted by the combination of the sign (with $N_b^s=N_b$) and amplitude ANNs (with $N_b^a=N_b/2$). For (c)–(f), the system size $L=24$.

Figure 4

(a) The structure of the ANN we used in the variational method calculations. (b) The “time” evolution of the excess energy per site $\mathrm{\Delta }e=(E_v-E_0)/L$ during the training process for a 1D $J_1\text{−}{J}_2$ AF Heisenberg model with $L=30$ using the trial wave function as shown in (a) with the neuron number in the hidden layer of the amplitude ANN $N_b=30$.