Neural network evolution strategy for solving quantum sign structures

Feed-forward neural networks are a novel class of variational wave functions for correlated many-body quantum systems. Here, we propose a specific neural network ansatz suitable for systems with real-valued wave functions. Its characteristic is to encode the all-important rugged sign structure of a quantum wave function in a convolutional neural network with discrete output. Its training is achieved through an evolutionary algorithm. We test our variational ansatz and training strategy on two spin-1/2 Heisenberg models, one on the two-dimensional square lattice and one on the three-dimensional pyrochlore lattice. In the former, our ansatz converges with high accuracy to the analytically known sign structures of ordered phases. In the latter, where such sign structures are a priori unknown, we obtain better variational energies than with other neural network states. Our results demonstrate the utility of discrete neural networks to solve quantum many-body problems.

Figure 1

Architecture of the helper network. The network has a convolutional layer followed by an activation function which returns a phase $\varphi$. One convolutional layer contains only one channel without bias. The red vector represents the sum of all blue vectors, and the green vectors are the outputs of phase and sign networks.

Figure 2

Comparison between gradient descent and evolution strategy (ES). Both of them are applicable in continuous functions, but only ES is applicable in the discrete case.

Figure 3

Cumulative distribution function (CDF) of phase relative to the MSR on the $10\times 10$ square lattice. The sign network learns all signs correctly in ${10}^6$ samples, resulting in the steplike CDF with the step at $\mathrm{\Phi }-{\mathrm{\Phi }}_{\mathrm{MSR}}=0$.

Figure 4

Relative error of the variational energy for the spin-1/2 $j_1\text{−}{j}_2$ Heisenberg model on the $6\times 6$ square lattice. The “base” and “base $+$ sign” points represent the performance of the base network without and with the sign network, respectively. The “CNN” data represent the approach employed in Ref. [11] which uses a complex-valued CNN to express the amplitude and phase with the MSR applied analytically. The energy on larger lattices (e.g., $10\times 10$) is not shown because the base network cannot approximate the ground state without the sign network in this case. The insets show the weights of the sign network at $j_2/j_1=0.5$ and $j_2/j_1=0.6$ on the $6\times 6$ and $10\times 10$ lattices.

Figure 5

Comparison of training efficiency between SR and ES. (a) Training time. (b) Accurate convergence probability. (c) Expected total time. The data for all panels are from the same training results on a Tesla-P100 GPU. In (a), every scattered point represents a training that converges within 40 s. In (c), the expected total time $t$ is given by Eq. (8).

Figure 6

Energy difference with respect to the ES approach in the pyrochlore Heisenberg model at various lattice sizes. The inset shows the infinite-size extrapolation for the NQS (SR) data. The NQS (SR) approach in Ref. [25] contains two separate CNNs for amplitude and phase. The many-variable variational Monte Carlo (mVMC) and ED data are also given in Ref. [25]. The DMRG data are given in Ref. [28].