Simulating disordered quantum Ising chains via dense and sparse restricted Boltzmann machines

S. Pilati and P. Pieri

Phys. Rev. E 101, 063308 – Published 22 June 2020

Abstract

In recent years, generative artificial neural networks based on restricted Boltzmann machines (RBMs) have been successfully employed as accurate and flexible variational wave functions for clean quantum many-body systems. In this article, we explore their use in simulations of disordered quantum Ising chains. The standard dense RBM with all-to-all interlayer connectivity is not particularly appropriate for large disordered systems, since in such systems one cannot exploit translational invariance to reduce the amount of parameters to be optimized. To circumvent this problem, we implement sparse RBMs, whereby the visible spins are connected only to a subset of local hidden neurons, thus reducing the amount of parameters. We assess the performance of sparse RBMs as a function of the range of the allowed connections, and we compare it with that of dense RBMs. Benchmark results are provided for two sign-problem-free Hamiltonians, namely pure and random quantum Ising chains. The RBM Ansätzes are trained using the unsupervised learning scheme based on projective quantum Monte Carlo (PQMC) algorithms. We find that the sparse connectivity facilitates the training process and allows sparse RBMs to outperform their dense counterparts. Furthermore, the use of sparse RBMs as guiding functions for PQMC simulations allows us to perform PQMC simulations at a reduced computational cost, avoiding possible biases due to finite random-walker populations. We obtain unbiased predictions for the ground-state energies and the magnetization profiles with fixed boundary conditions, at the ferromagnetic quantum critical point. The magnetization profiles agree with the Fisher–de Gennes scaling relation for conformally invariant systems, including the scaling dimension predicted by the renormalization-group analysis.

Received 25 March 2020
Accepted 2 June 2020

DOI:https://doi.org/10.1103/PhysRevE.101.063308

Physics Subject Headings (PhySH)

1-dimensional spin chains

Machine learning Monte Carlo methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

S. Pilati ¹ and P. Pieri^1,2

¹School of Science and Technology, Physics Division, Università di Camerino, 62032 Camerino (MC), Italy
²INFN, Sezione di Perugia, 06123 Perugia (PG), Italy

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Vol. 101, Iss. 6 — June 2020

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Images

Figure 1
Representation of the connectivity structures of four different artificial neural networks. Segments indicate the allowed interactions. Diagrams (a) and (b) represent examples of the sparse restricted Boltzmann machine employed in this article, whereby visible spins interact with a limited number of local hidden neurons; in the cases visualized here, this number is $N_{c} = 2$ (a) and $N_{c} = 3$ (b). Diagram (c) represents the standard dense restricted Boltzmann machine with all-to-all inter-layer connectivity [1]. Diagram (d) represents a type of unrestricted Boltzmann machine [17], alias a shadow wave function [57, 58]. This is characterized by interlayer connections only among spins corresponding to the same index, plus nearest-neighbor intralayer connections.
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Figure 2
Energy per spin $E / N$ corresponding to two optimized neural-network Ansätzes as a function of the number of connected hidden spins $N_{c}$ . For the dense RBMs, which are represented by the empty symbols, $N_{c}$ coincides with the number of hidden spins $N_{h}$ . The full symbols correspond to the sparse RBMs, for which $N_{c}$ is the number of allowed local connections. The (blue) squares represent the results for the clean ferromagnetic Ising model, and they refer to the left vertical axis. The (brown) circles correspond to one realization of the random Ising model with couplings sampled from the flat distribution (see text), and they refer to the right vertical axis. Both models include $N = 80$ spins and they are simulated at the coupling strength corresponding to the respective ferromagnetic quantum critical points. The horizontal lines indicate the exact ground-state energies computed through the Jordan-Wigner transformation for $N = 80$ .
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Figure 3
Histogram of the relative errors $e_{rel} = (E - E_{JW}) / |E_{JW}|$ of the ground-state energies $E$ corresponding to the optimized RBM Ansätzes, with respect to the (exact) Jordan-Wigner result $E_{JW}$ . A total of 60 realizations of the random couplings are considered. The transverse field intensity is set at the ferromagnetic quantum critical point. The full columns correspond to three sparse RBMs different connection numbers $N_{c}$ . The empty columns correspond to a dense RBM with $N_{h} = 80$ hidden neurons. The width of some columns is reduced for visibility.
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Figure 4
Histogram of the relative errors $e_{rel}$ of the ground-state energies $E$ predicted by projective quantum Monte Carlo simulations guided by optimized sparse RBMs. A total of 60 realizations of the random couplings of the quantum Ising chain are considered. The system parameters are as in Fig. 3. The three datasets correspond to different connection numbers $N_{c}$ . The width of some columns is reduced for visibility.
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Figure 5
Profile of the rescaled local magnetization $N^{χ} m (j)$ as a function of the rescaled spin index $j' = (j - 1 / 2) / N$ , where $j = 1, \dots, N$ is the site index and $N$ is the system size. $m (j)$ is the average magnetization at the site $j$ , and $χ$ is the scaling dimension of the magnetization. The different symbols represent the PQMC predictions for different $N$ . The system has fixed boundary conditions $m (1) = m (N) = 1$ . The transverse field intensity corresponds to the ferromagnetic quantum critical point. Panel (a) corresponds to a pure ferromagnetic Ising model. Panel (b) corresponds to a random Ising chain, with the couplings sampled from a binary distribution (see the text). The results are averaged over 1200 realizations. The continuous (black) curves represent fitting functions based on the Fisher–de Gennes scaling relation for conformally invariant models; see Eq. (9). The fitted scaling dimensions are consistent with the expected results, $x_{m} = 1 / 8$ and $x_{m} = (3 - \sqrt{5}) / 4$ , for the pure and the random models, respectively (see the text).
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