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arXiv:2005.14290 · Physical review. E

Scalable neural networks for the efficient learning of disordered quantum systems

N. Saraceni, Simone Cantori, Sebastiano Pilati

Published Sep 1, 2020• 21 citations
arXivHTMLPDF

Abstract

Supervised machine learning is emerging as a powerful computational tool to predict the properties of complex quantum systems at a limited computational cost. In this article, we quantify how accurately deep neural networks can learn the properties of disordered quantum systems as a function of the system size. We implement a scalable convolutional network that can address arbitrary system sizes. This network is compared with a recently introduced extensive convolutional architecture [Mills et al., Chem. Sci. 10, 4129 (2019)2041-652010.1039/C8SC04578J] and with conventional dense networks with all-to-all connectivity. The networks are trained to predict the exact ground-state energies of various disordered systems, namely, a continuous-space single-particle Hamiltonian for cold-atoms in speckle disorder, and different setups of a quantum Ising chain with random couplings, including one with only short-range interactions and one augmented with a long-range term. In all testbeds we consider, the scalable network retains high accuracy as the system size increases. Furthermore, we demonstrate that the network scalability enables a transfer-learning protocol, whereby a pretraining performed on small systems drastically accelerates the learning of large-system properties, allowing reaching high accuracy with small training sets. In fact, with the scalable network one can even extrapolate to sizes larger than those included in the training set, accurately reproducing the results of state-of-the-art quantum Monte Carlo simulations.

DOI
10.1103/physreve.102.033301

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Supervised machine learning is emerging as a powerful computational tool to predict the properties of complex quantum systems at a limited computational cost. In this article, we quantify how accurately deep neural networks can learn the properties of disordered quantum systems as a function of the system size. We implement a scalable convolutional network that can address arbitrary system sizes. This network is compared with a recently introduced extensive convolutional architecture [Mills et al., Chem. Sci. 10, 4129 (2019)2041-652010.1039/C8SC04578J] and with conventional dense networks with all-to-all connectivity. The networks are trained to predict the exact ground-state energies of various disordered systems, namely, a continuous-space single-particle Hamiltonian for cold-atoms in speckle disorder, and different setups of a quantum Ising chain with random couplings, including one with only short-range interactions and one augmented with a long-range term. In all testbeds we consider, the scalable network retains high accuracy as the system size increases. Furthermore, we demonstrate that the network scalability enables a transfer-learning protocol, whereby a pretraining performed on small systems drastically accelerates the learning of large-system properties, allowing reaching high accuracy with small training sets. In fact, with the scalable network one can even extrapolate to sizes larger than those included in the training set, accurately reproducing the results of state-of-the-art quantum Monte Carlo simulations.
Neural-network quantum states have shown great potential for the study of many-body quantum systems. In statistical machine learning, transfer learning designates protocols reusing features of a machine learning model trained for a problem to solve a possibly related but different problem. We propose to evaluate the potential of transfer learning to improve the scalability of neural-network quantum states. We devise and present physics-inspired transfer learning protocols, reusing the features of neural-network quantum states learned for the computation of the ground state of a small system for systems of larger sizes. We implement different protocols for restricted Boltzmann machines on general-purpose graphics processing units. This implementation alone yields a speedup over existing implementations on multicore and distributed central processing units in comparable settings. We empirically and comparatively evaluate the efficiency (time) and effectiveness (accuracy) of different transfer learning protocols as we scale the system size in different models and different quantum phases. Namely, we consider both the transverse field Ising and Heisenberg XXZ models in one dimension, as well as in two dimensions for the latter, with system sizes up to 128 and 8×8 spins. We empirically demonstrate that some of the transfer learning protocols that we have devised can be far more effective and efficient than starting from neural-network quantum states with randomly initialized parameters.
Abstract Predicting the output of quantum circuits is a hard computational task that plays a pivotal role in the development of universal quantum computers. Here we investigate the supervised learning of output expectation values of random quantum circuits. Deep convolutional neural networks (CNNs) are trained to predict single-qubit and two-qubit expectation values using databases of classically simulated circuits. These circuits are built using either a universal gate set or a continuous set of rotations plus an entangling gate, and they are represented via properly designed encodings of these gates. The prediction accuracy for previously unseen circuits is analyzed, also making comparisons with small-scale quantum computers available from the free IBM Quantum program. The CNNs often outperform these quantum devices, depending on the circuit depth, on the network depth, and on the training set size. Notably, our CNNs are designed to be scalable. This allows us exploiting transfer learning and performing extrapolations to circuits larger than those included in the training set. These CNNs also demonstrate remarkable resilience against noise, namely, they remain accurate even when trained on (simulated) expectation values averaged over very few measurements.
Density functional theory (DFT) is routinely employed in material science and quantum chemistry to simulate weakly correlated electronic systems. Recently, deep learning (DL) techniques have been adopted to develop promising functionals for the strongly correlated regime. DFT can be applied to quantum spin models too, but functionals based on DL have not been developed yet. Here, we investigate DL-based DFTs for random quantum Ising chains, both with nearest-neighbor and up to next-nearest-neighbor couplings. Our neural functionals are trained and tested on data produced via the Jordan-Wigner transformation, exact diagonalization, and tensor-network methods. An economical gradient-descent optimization is used to find the ground-state properties of previously unseen Hamiltonian instances. Notably, our nonlocal functionals drastically improve upon the common local density approximations, and they are designed to be scalable, allowing us to simulate chain sizes much larger than those used for training. The prediction accuracy is analyzed, paying attention to the spatial correlations of the learned functionals and to the role of quantum criticality. Our findings indicate a suitable strategy to extend the reach of other computational methods with a controllable accuracy.
Over the past years, machine learning has emerged as a powerful computational tool to tackle complex problems over a broad range of scientific disciplines. In particular, artificial neural networks have been successfully deployed to mitigate the exponential complexity often encountered in quantum many-body physics, the study of properties of quantum systems built out of a large number of interacting particles. In this Article, we overview some applications of machine learning in condensed matter physics and quantum information, with particular emphasis on hands-on tutorials serving as a quick-start for a newcomer to the field. We present supervised machine learning with convolutional neural networks to learn a phase transition, unsupervised learning with restricted Boltzmann machines to perform quantum tomography, and variational Monte Carlo with recurrent neural-networks for approximating the ground state of a many-body Hamiltonian. We briefly review the key ingredients of each algorithm and their corresponding neural-network implementation, and show numerical experiments for a system of interacting Rydberg atoms in two dimensions.
Over the past few years, machine learning has emerged as a powerful computational tool to tackle complex problems in a broad range of scientific disciplines.In particular, artificial neural networks have been successfully used to mitigate the exponential complexity often encountered in quantum many-body physics, the study of properties of quantum systems built from a large number of interacting particles.In this article, we review some applications of neural networks in condensed matter physics and quantum information, with particular emphasis on hands-on tutorials serving as a quick start for a newcomer to the field.The prerequisites of this tutorial are basic probability theory and calculus, linear algebra, basic notions of neural networks, statistical physics, and quantum mechanics.The reader is introduced to supervised machine learning with convolutional neural networks to learn a phase transition, unsupervised learning with restricted Boltzmann machines to perform quantum tomography, and the variational Monte Carlo method with recurrent neural networks for approximating the ground state of a many-body Hamiltonian.For each algorithm, we briefly review the key ingredients and their corresponding neuralnetwork implementation, and show numerical experiments for a system of interacting Rydberg atoms in two dimensions.
We are interested in how quantum data can allow for practical solutions to otherwise difficult computational problems. A notoriously difficult phenomenon from quantum many-body physics is the emergence of many-body localization (MBL). So far, is has evaded a comprehensive analysis. In particular, numerical studies are challenged by the exponential growth of the Hilbert space dimension. As many of these studies rely on exact diagonalization of the system's Hamiltonian, only small system sizes are accessible. In this work, we propose a highly flexible neural network based learning approach that, once given training data, circumvents any computationally expensive step. In this way, we can efficiently estimate common indicators of MBL such as the adjacent gap ratio or entropic quantities. Our estimator can be trained on data from various system sizes at once which grants the ability to extrapolate from smaller to larger ones. Moreover, using transfer learning we show that already a two-dimensional feature vector is sufficient to obtain several different indicators at various energy densities at once. We hope that our approach can be applied to large-scale quantum experiments to provide new insights into quantum many-body physics.