Restricted Boltzmann machine learning for solving strongly correlated quantum systems

We develop a machine learning method to construct accurate ground-state wave functions of strongly interacting and entangled quantum spin as well as fermionic models on lattices. A restricted Boltzmann machine algorithm in the form of an artificial neural network is combined with a conventional variational Monte Carlo method with pair product (geminal) wave functions and quantum number projections. The combination allows an application of the machine learning scheme to interacting fermionic systems. The combined method substantially improves the accuracy beyond that ever achieved by each method separately, in the Heisenberg as well as Hubbard models on square lattices, thus proving its power as a highly accurate quantum many-body solver.

Figure 1

Schematic illustration of (a) P-RBM and (b) RBM+PP to represent many-body wave function. Physical variables in the visible layer couple to artificial neurons in the hidden layer through $W_{ik}$ interactions in Eq. (2). Whereas no entanglement among physical variables exists in the absence of the hidden layer in P-RBM, RBM+PP provides direct entanglement via $f_{ij}^{\sigma {\sigma }^{\prime }}$ parameters in Eq. (3). For visibility, only a small portion of connections by $W_{ik}$ and $f_{ij}^{\sigma {\sigma }^{\prime }}$ are shown.

Figure 2

RBM+PP results for energy of 2D AFM Heisenberg model defined on the $8\times 8$ square lattice with fully periodic boundary condition. (a) Relative error of energy to SSE-QMC energy ($E/J=-0.673487\left(4\right)$) [61] as a function of $1/\alpha$ ($\alpha$: hidden variable density). $△$ ($▽$) symbol: RBM+PP $|\mathrm{\Psi }\rangle ={\mathcal{NL}}^{K=0}{\mathcal{L}}^{S=0}{\mathcal{P}}_{\mathrm{G}}^{\infty }|{\varphi }_{\mathrm{pair}}\rangle$ ($|\mathrm{\Psi }\rangle ={\mathcal{NL}}^{S=0}{\mathcal{P}}_{\mathrm{G}}^{\infty }|{\varphi }_{\mathrm{pair}}\rangle$). $◯$ symbol: P-RBM $|\mathrm{\Psi }\rangle =\mathcal{N}|{\varphi }_{\mathrm{product}}\rangle$. $\alpha =0$ (solid horizontal lines) corresponds to the mVMC results (red: $|\mathrm{\Psi }\rangle ={\mathcal{L}}^{K=0}{\mathcal{L}}^{S=0}{\mathcal{P}}_{\mathrm{G}}^{\infty }|{\varphi }_{\mathrm{pair}}\rangle$, green: $|\mathrm{\Psi }\rangle ={\mathcal{L}}^{S=0}{\mathcal{P}}_{\mathrm{G}}^{\infty }|{\varphi }_{\mathrm{pair}}\rangle$). Two arrows (from top to bottom) indicate the result of entangled-plaquette states (EPS) [62] and variational QMC to evaluate the projected entangled pair states (PEPS) for virtual bond dimensions of 16 [63]. (b) Variance ${\mathrm{\Delta }}_{\mathrm{var}}$ extrapolation of energy. The cross ($\times$) on the ordinate shows the SSE-QMC energy. The data points plotted in this variance range are $\alpha =0$, 2, 4, 8, 16, 32 (from right to left) for red $△$ symbols, $\alpha =2$, 4, 8, 16, 32 for green $▽$ symbols, and $\alpha =8$, 16, 32 for blue $◯$ symbols, respectively. Linear fit and extrapolation to ${\mathrm{\Delta }}_{\mathrm{var}}\to 0$ is shown as solid lines. All the data points in this range except $\alpha =0$ data in red are used in the fit. Error bars show standard errors of Monte Carlo measurements of energy [and also variance in case of (b)] for the optimized variational wave function.

Figure 3

RBM+PP result for energy of 2D Hubbard model at half filling with (a) $U/t=4$ and (b) $U/t=8$ on $8\times 8$ square lattice with P-AP boundary condition. Relative error of the RBM+PP energy to the AF-QMC energy ($E/t=-0.8642\left(2\right)$ and $-0.5259\left(3\right)$ for $U/t=4$ and $U/t=8$, respectively) [69] as a function of $1/\alpha$ ($\alpha$: hidden variable density) is shown. $△$ ($▽$) symbol: RBM+PP $|\mathrm{\Psi }\rangle ={\mathcal{NL}}^{K=0}|{\varphi }_{\mathrm{pair}}\rangle$ ($|\mathrm{\Psi }\rangle =\mathcal{N}|{\varphi }_{\mathrm{pair}}\rangle$). $◯$ symbol: F-RBM $|\mathrm{\Psi }\rangle =\mathcal{N}|{\varphi }_{\mathrm{Fermi}\text{−}\mathrm{sea}}\rangle$. Solid horizontal red (green) lines: results of mVMC functions $|\mathrm{\Psi }\rangle ={\mathcal{P}}_{\mathrm{G}}{\mathcal{P}}_{\mathrm{J}}{\mathcal{L}}^{K=0}|{\varphi }_{\mathrm{pair}}\rangle$ ($|\mathrm{\Psi }\rangle ={\mathcal{P}}_{\mathrm{G}}{\mathcal{P}}_{\mathrm{J}}|{\varphi }_{\mathrm{pair}}\rangle$). The arrow in (a) indicates the TNVMC result [70]. Error bars show standard errors of Monte Carlo measurements of energy for the optimized variational wave function.

Figure 4

$U$ dependence of RBM+PP (${\mathcal{NL}}^{K=0}|{\varphi }_{\mathrm{pair}}\rangle$ with $\alpha =32$) energy of 2D Hubbard model defined on the $8\times 8$ square lattice with P-AP boundary condition. Relative error of the RBM+PP energy to that obtained by the AF-QMC [69, 71] is shown. For comparison, at $t/U=0$, we show the RBM+PP (${\mathcal{NL}}^{K=0}{\mathcal{L}}^{S=0}{\mathcal{P}}_{\mathrm{G}}^{\infty }|{\varphi }_{\mathrm{pair}}\rangle$ with $\alpha =32$) result for the $8\times 8$ Heisenberg model. Error bars show standard errors of Monte Carlo measurements of energy for the optimized variational wave function.