The eigenvalue problem of quantum many-body systems is a fundamental and challenging subject in condensed-matter physics because the dimension of the Hilbert space (and hence the required computational memory and time) grows exponentially as the system size increases. A few numerical methods have been developed for some specific systems but may not be applicable in others. Here we propose a general numerical method, random sampling neural networks (RSNNs), to utilize the pattern recognition technique for the random sampling matrix elements of an interacting many-body system via a self-supervised learning approach. Several exactly solvable one-dimensional models, including the Ising model with a transverse field, the Fermi-Hubbard model, and the spin-
21XXZ model, are used to test the applicability of RSNN. Pretty high accuracy (
>96) of energy spectra, magnetization, critical exponents, etc. can be obtained within the strongly correlated regime or near the quantum phase-transition point, even the corresponding RSNN models are trained in the weakly interacting regime. After including data augmentation and transfer learning methods, RSNN can be further applied to systems of larger sizes with much less training data if only pretrained in a smaller system. Our results demonstrate the possibility to combine the existing numerical methods and the RSNN to explore quantum many-body problems in a much wider parameter space.
Neural network approaches to machine learning problems rely on statistical frameworks that are designed to generalize to many situations, whereas traditional physics-based approaches to quantum problems stem from specific physical laws that govern the behavior of certain classes of systems. While quantum physics and machine learning have been largely disparate fields for most of their history, their interplay has gained considerable attention in recent years, primarily driven by the premise that the most powerful computational models are those whose foundations stem from both physical and statistical principles. Two overarching strategies in this area include building physics-based machine learning models from the ground up, or alternatively, looking for hidden ways that statistical models include aspects of physical systems. Many-body quantum systems turn out to be a useful testing ground for both of these approaches, due to the complex behavior that arises from simple fundamental rules. Here, we present several ways that neural networks can represent many-body spin states, which enables the exploration into the underlying nature of neural networks, as well as gaining insight into how effective different classes of models are for simulating physical systems. The main focus areas of this thesis are (1) connecting convolutional neural networks to maximum-entropy distributions when applied to spin systems, (2) using generative models to gain physical insights into experimental tomographic data, and (3) applying inverse reinforcement learning to probe the nature of agent-based quantum circuit design.
A fundamental problem in quantum many-body physics is that of finding ground states of local Hamiltonians. A number of recent works gave provably efficient machine learning (ML) algorithms for learning ground states. Specifically, [Huang et al. Science 2022], introduced an approach for learning properties of the ground state of an
n-qubit gapped local Hamiltonian
H from only
nO(1) data points sampled from Hamiltonians in the same phase of matter. This was subsequently improved by [Lewis et al. Nature Communications 2024], to
O(logn) samples when the geometry of the
n-qubit system is known. In this work, we introduce two approaches that achieve a constant sample complexity, independent of system size
n, for learning ground state properties. Our first algorithm consists of a simple modification of the ML model used by Lewis et al. and applies to a property of interest known beforehand. Our second algorithm, which applies even if a description of the property is not known, is a deep neural network model. While empirical results showing the performance of neural networks have been demonstrated, to our knowledge, this is the first rigorous sample complexity bound on a neural network model for predicting ground state properties. We also perform numerical experiments that confirm the improved scaling of our approach compared to earlier results.
In this paper we discuss in detail the
P(I)′s, angular momentum I probabilities in the ground states, of many-body systems interacting via a two-body random ensemble (TBRE). In particular, we extensively apply an approach introduced in an earlier paper and compare the predicted
P(I)′s with those obtained by diagonalizing a TBRE Hamiltonian. We begin with a few solvable cases, such as fermions in a small single-
j shell and d boson systems, where elegant agreements between the predicted
P(I)′s and those obtained by diagonalizing a TBRE Hamiltonian are achieved. We find that d boson systems systematically present counterexamples of angular momentum 0 ground state dominance when the number of d bosons is $6\ensuremath{\kappa}\ifmmode\pm\else\textpm\fi{}1$ with $\ensuremath{\kappa}$ a natural number, which suggests that certain fundamental symmetry (say, time reversal invariance) of the Hamiltonian cannot ensure the occurrence of angular momentum 0 ground state dominance. Next, we apply the same approach to more complicated cases, such as even or odd number of fermions in a large single-
j shell or a many-
j shell,
sd-boson or
sdg-boson systems, etc. We find that the simple approach proposed in an earlier paper is also well applicable, and thus it is a universal approach. The numerical experiments provide a guideline to tell which interactions are essential to produce a sizable
P(I) in a many-body system. This disproves a popular idea that the angular momentum 0 ground state (0 g.s.) dominance may be independent of two-body interactions. Some matrix elements which are useful to understand the observed regularities are given or addressed in detail. In this paper we also report a synchronous staggering between the 0 g.s. probabilities of even numbers of fermions in a single-
j shells and j g.s. probabilities of odd numbers of fermions in a single-
j shell when j is small. The low seniority chain of 0 g.s. using the same set of two-body interactions is confirmed, but it is noted that contribution to the total 0 g.s. probability beyond this chain may be more important for even numbers of fermions in a single-
j shell. Some interesting results taking a displaced two-body random ensemble are presented for the I ground state probabilities.
Most interacting many-body systems in physics are not analytically solvable. Instead, numerical methods are needed for the study of these complex and high-dimensional problems. At present, there are many interesting problems in strongly correlated systems that remain unsolved with current methods. At the heart of this problem is finding an efficient representation that incorporates symmetries, correlations and general features. \n \nIn the context of computer science, machine learning techniques have had astonishing success at reducing the dimensionality of data. The leading method is through the use of artificial neural networks. These networks have been enormously successful at sifting through vast amounts of data to find patterns and regularities. In a sense, neural networks are themselves a statistical system whose properties are adjusted to mimic the features of the data. By finding an effective low-dimensional representation of the data, machine learning has greatly subdued the curse of dimensionality found in many real-world problems. \n \nIn this Thesis, we apply several machine learning techniques to the study of interacting many-body systems in classical and quantum statistical physics. We explore supervised classification of phases of matter with an emphasis on physical interpretation of the net- work. In doing so, we design a custom network architecture that possesses rotational symmetry as an inductive bias. We further investigate connections between the renormalization group and deep learning through applying a super-resolving neural network to the classical Ising model. Towards experimental efforts, we also repurpose generative machine learning to quantum state tomography for the calibration and testing of quantum devices. We conclude with a latent variable model inspired by near-term quantum algorithms. This maps to a variational Monte Carlo ansatz that produces samples efficiently for interacting quantum systems.