Self-learning projective quantum Monte Carlo simulations guided by restricted Boltzmann machines

The projective quantum Monte Carlo (PQMC) algorithms are among the most powerful computational techniques to simulate the ground-state properties of quantum many-body systems. However, they are efficient only if a sufficiently accurate trial wave function is used to guide the simulation. In the standard approach, this guiding wave function is obtained in a separate simulation that performs a variational minimization. Here we show how to perform PQMC simulations guided by an adaptive wave function based on a restricted Boltzmann machine. This adaptive wave function is optimized along the PQMC simulation via unsupervised machine learning, avoiding the need of a separate variational optimization. As a byproduct, this technique provides an accurate ansatz for the ground-state wave function, which is obtained by minimizing the Kullback-Leibler divergence with respect to the PQMC samples, rather than by minimizing the energy expectation value as in standard variational optimizations. The high accuracy of this self-learning PQMC technique is demonstrated for a paradigmatic sign-problem-free model, namely, the ferromagnetic quantum Ising chain, showing very precise agreement with the predictions of the Jordan-Wigner theory and of loop quantum Monte Carlo simulations performed in the low-temperature limit.

Figure 1

Connectivity structure of the restricted Boltzmann machine (RBM) [23] employed in this work as PQMC guiding wave function (upper panel) and of the unrestricted Boltzmann machine employed in Ref. [24] (lower panel). The latter is analogous to the shadow wave function used for quantum fluids and solids [25, 26]. The visible spins are labeled by $x_j$, with $j=1,\cdots ,N$, while the hidden spins by $h_i$, with $i=1,\cdots ,N_h$. The segments indicate the allowed interactions.

Figure 2

Self-learning PQMC simulation of a ferromagnetic quantum Ising chain with $N=80$ spins and open boundary conditions. The transverse-field intensity is $\mathrm{\Gamma }=0.85$, the longitudinal-field intensity is $h=0$. The energy per spin $E/N$ is plotted as a function of the imaginary time $\tau$. The (violet) dashed curve indicates the running average in a simple PQMC simulation performed without a guiding wave function. The (dark-green) dot-dashed curve corresponds to a PQMC simulation guided by a fixed RBM wave function ${\psi }_{G_{s=0}}\left(\mathit{x}\right)$ with $N_h=20$ hidden spins and random parameters. The (red) continuous curve corresponds to the PQMC simulation guided by the adaptive RBM wave function ${\psi }_{G_s}\left(\mathit{x}\right)$ trained with unsupervised learning. The (blue) squares indicate the average energy $E_{{\psi }_{G_s}}$ corresponding to the adaptive RBM wave function ${\psi }_{G_s}\left(\mathit{x}\right)$ obtained after each stint $s$. The (brown) horizontal bar indicates the minimal variational energy $E_{\mathrm{var}.\min .}$ for $N_h=20$ obtained with the NetKet library [47] using the stochastic reconfiguration algorithm. The (black) horizontal line corresponds to the exact ground-state energy computed via Jordan-Wigner transformation [67].

Figure 3

Relative error $(E-E_{\mathrm{JW}})/\left|E_{\mathrm{JW}}\right|$ of the estimated ground-state energy $E$ with respect to the (exact) Jordan-Wigner result $E_{\mathrm{JW}}$, as a function of the hidden-spin number $N_h$. The system parameters are: $N=80$, $\mathrm{\Gamma }=1$, and $h=0$. The (red) squares indicate the energy expectation value of the optimal adaptive RBM wave-function ${\psi }_{G_{s\to \infty }}\left(\mathit{x}\right)$, optimized via unsupervised learning along the PQMC simulation. This optimization corresponds to the minimization of the Kullback-Leibler divergence. The (blue) circles correspond to the minimal variational energy for an RBM wave-function obtained with the NetKet library using the stochastic reconfiguration algorithm. As a reference, the minimal variational energy corresponding to the unrestricted Boltzmann machine (uRBM) wave function [24] is indicated by the horizontal (green) bar. The width represents (twice) the statistical error. The uRBM wave function is optimized using the stochastic gradient descent algorithm.

Figure 4

Energy per spin $E/N$ obtained from a PQMC simulation guided by the optimized adaptive RBM wave functions ${\psi }_{G_{s\to \infty }}\left(\mathit{x}\right)$ with different numbers of hidden spins $N_h$. The system parameters are: $N=80$, $\mathrm{\Gamma }=1$, and $h=0$. The (blue) horizontal line indicates the exact ground-state energy computed via the Jordan-Wigner (JW) transformation. For comparison, the (green) dot-dashed line indicates the results of a PQMC simulation guided by an optimized Boltzmann-type ansatz.

Figure 5

Main panel: energy per spin $E/N$ as a function of the system size $N$, at the quantum critical point $\mathrm{\Gamma }=1$ with $h=0$. The (red) dots and interpolating line indicate the exact Jordan-Wigner result. The (blue) circles indicate the results of PQMC simulation guided by the optimized adaptive RBM wave function ${\psi }_{G_{s\to \infty }}\left(\mathit{x}\right)$ with $N_h=N/2$ hidden spins. Inset: ratio of the PQMC result $E_{\mathrm{PQMC}}$ and the exact (Jordan-Wigner) ground-state energy $E_{\mathrm{JW}}$.

Figure 6

Energy per spin $E/N$ as a function of the transverse magnetic field $\mathrm{\Gamma }$, for vanishing longitudinal magnetic field, i.e., $h=0$, and for $h=0.2$. The system size is $N=80$. At $h=0$, the results of PQMC simulations guided by the optimized adaptive RBM wave function ${\psi }_{G_{s\to \infty }}\left(\mathit{x}\right)$ (blue squares) are compared with the Jordan-Wigner theory (red dots and interpolating line). At $h=0.2$, comparison is made with loop QMC simulations performed at low temperature $T=0.05$ using the ALPS library [44].

Figure 7

Main panel: magnetization per spin $M/N$ as a function of the transverse magnetic field $\mathrm{\Gamma }$. The system size is $N=80$ and the longitudinal-field intensity is $h=0.2$. The PQMC results obtained with the pure estimator (blue empty circles) are compared with the results of loop QMC simulations (green crosses) performed with the ALPS library at the temperature $T=0.05$. Inset: loop QMC results (green bullets) for $M/N$ as a function of the temperature $T$, for $\mathrm{\Gamma }=1$ and $h=0.2$. The (red) horizontal thin bar corresponds to the zero-temperature PQMC prediction obtained using the mixed estimator. The horizontal bar with (blue) diagonal pattern corresponds to the pure estimator.