Despite the many challenges in exploratory data analysis, artificial neural networks have motivated strong interests in scientists and researchers both in theoretical as well as practical applications. Among sources of such popularity of artificial neural networks the ability of modeling non-linear dynamical systems, generalization, and adaptation possibilities should be mentioned. Despite this, there is still significant debate about the role of various underlying stochastic processes in stabilizing a unique structure for data learning and prediction. One of such obstacles to the theoretical and numerical study of machine intelligent systems is the curse of dimensionality and the sampling from high-dimensional probability distributions. In general, this curse prevents efficient description of states, providing a significant complexity barrier for the system to be efficiently described and studied. In this strand of research, direct treatment and description of such abstract notions of learning theory in terms of quantum information be one of the most favorable candidates. Hence, the subject matter of these articles is devoted to problems of design, adaptation and the formulations of computationally hard problems in terms of quantum mechanical systems. In order to characterize the microscopic description of such dynamics in the language of inferential statistics, covariance matrix estimation of d-dimensional Gaussian densities and Bayesian interpretation of eigenvalue problem for dynamical systems is assessed.
Despite the many challenges in exploratory data analysis, artificial neural networks have motivated strong interests in scientists and researchers both in theoretical as well as practical applications. Among sources of such popularity of artificial neural networks the ability of modeling non-linear dynamical systems, generalization, and adaptation possibilities should be mentioned. Despite this, there is still significant debate about the role of various underlying stochastic processes in stabilizing a unique structure for data learning and prediction. One of such obstacles to the theoretical and numerical study of machine intelligent systems is the curse of dimensionality and the sampling from high-dimensional probability distributions. In general, this curse prevents efficient description of states, providing a significant complexity barrier for the system to be efficiently described and studied. In this strand of research, direct treatment and description of such abstract notions of learning theory in terms of quantum information be one of the most favorable candidates. Hence, the subject matter of these articles is devoted to problems of design, adaptation and the formulations of computationally hard problems in terms of quantum mechanical systems. In order to characterize the microscopic description of such dynamics in the language of inferential statistics, covariance matrix estimation of d-dimensional Gaussian densities and Bayesian interpretation of eigenvalue problem for dynamical systems is assessed.
The simulation of quantum matter with classical hardware plays a central role in the discovery and development of quantum many-body systems, with far-reaching implications in condensed matter physics and quantum technologies. In general, efficient and sophisticated algorithms are required to overcome the severe challenge posed by the exponential scaling of the Hilbert space of quantum systems. In contrast, hardware built with quantum bits of information are inherently capable of efficiently finding solutions of quantum many-body problems. While a universal and scalable quantum computer is still beyond the horizon, recent advances in qubit manufacturing and coherent control of synthetic quantum matter are leading to a new generation of intermediate scale quantum hardware. \n \nThe complexity underlying quantum many-body systems closely resembles the one encountered in many problems in the world of information and technology. In both contexts, the complexity stems from a large number of interacting degrees of freedom. A powerful strategy in the latter scenario is machine learning, a subfield of artificial intelligence where large amounts of data are used to extract relevant features and patterns. In particular, artificial neural networks have been demonstrated to be capable of discovering low-dimensional representations of complex objects from high-dimensional dataset, leading to the profound technological revolution we all witness in our daily life. \n \nIn this Thesis, we envision a new paradigm for scientific discovery in quantum physics. On the one hand, we have the essentially unlimited data generated with the increasing amount of highly controllable quantum hardware. On the other hand, we have a set of powerful algorithms that efficiently capture non-trivial correlations from high-dimensional data. Therefore, we fully embrace this data-driven approach to quantum mechanics, and anticipate new exciting possibilities in the field of quantum many-body physics and quantum information science. We revive a powerful stochastic neural network called a restricted Boltzmann machine, which slowly moved out of fashion after playing a central role in the machine learning revolution of the early 2010s. We introduce a neural-network representation of quantum states based on this generative model. We propose a set of algorithms to reconstruct unknown quantum states from measurement data and numerically demonstrate their potential, with important implications for current experiments. These include the reconstruction of experimentally inaccessible properties, such as entanglement, and diagnostics to determine sources of noise. Furthermore, we introduce a machine learning framework for quantum error correction, where a neural network learns the best decoding strategy directly from data. We expect that the full integration between quantum hardware and artificial intelligence will become the gold standard, and will drive the world into the era of fault-tolerant quantum computing and large-scale quantum simulations.
One of the most promising new directions in quantum computing and artificial intelligence convergence is the Quantum Neural Networks (QNNs) that can potentially revolutionize the manner in which intelligent systems are designed, trained, and implemented. Although traditional deep learning models have proven to be incredibly successful in fields like computer vision, natural language processing and autonomous systems, they are becoming limited by computational bottlenecks, energy use, and scalability constraints when operating on large high-dimensional data. The distinctive principles of quantum mechanics (superposition, entanglement, and quantum parallelism) may be applied to the QNNs to provide a distinct computational paradigm with the ability to execute exponentially complicated computations more effectively than his classical counterparts. This article gives an in-depth analysis of QNNs as the new frontier of artificial intelligence. We start with the description of the theoretical foundations of quantum computation and their realization in the form of neural networks. We then examine state-of-the-art QNN architectures, such as hybrid quantum-classical models that help overcome present quantum hardware limitations in relation to the capabilities of classical machine learning. Some of the upcoming applications are showcased including molecular simulation and drug discovery and financial modeling, optimization, cryptography and intelligent automation. Such applications reveal the revolutionary power of QNNs to tackle the issues that cannot be tackled by traditional systems. Nonetheless, there are also major obstacles on the way to a practical implementation of QNN. Problems that still remain a barrier to widespread adoption include quantum decoherence, noise in near-term quantum devices, and the inability to scale qubit architectures or to train large-scale quantum models. In order to overcome these obstacles, we speak about the current research work on the topic of error correction, variational algorithms, and hybrid system designs. Also we examine the ethical and societal ramifications of quantum-accelerated AI, such as security, data sovereignty, and technological equity globally.
Quantum machine learning is at the crossroads of two of the most exciting current areas of research: quantum computing and classical machine learning. Although the field is still in its infancy, the body of literature is already large enough to warrant several review articles [1–3]. This short survey focuses on a selection of significant recent results on the subtopic of quantum neural networks, an area that hopes to build on the enormous impact that classical neural networks have had. In particular, we concentrate on quantum generalizations of popular neural-network concepts such as Boltzmann machines, generative adversarial networks and autoencoders, as well as applications of classical neural networks to quantum information and vice versa. Examples of Boltzmann machines. In a fully connected Boltzmann machine, connections exist between all visible and hidden neurons. In a restricted Boltzmann machine (RBM), there are no connections between neurons of the same type. A deep Boltzmann machine (DBM) consists of adding one or more additional layers of hidden neurons to an RBM, with no connections between neurons in the same layer. The energy-based nature of BMs gives a natural framework for considering quantum generalizations of their behavior. The BM energy function is equivalent to the Hamiltonian of a simple Ising model and one might hope that more general Hamiltonians allowed by quantum mechanics could explain certain data sets better than classically. Amin et al. [4] adopt this approach, and take the so-called transverse field Ising Hamiltonian as the energy function. This increased generality brings certain difficulties, however, as the transverse field term makes training the quantum Boltzmann machine (QBM) non-trivial. Although the general scaling and performance of QBMs is not yet well understood, for certain small-size examples the authors observe that their QBMs can learn data distributions better than classical BMs. Kieferova and Wiebe [5] extend the formalism of Amin et al. and consider more general Hamiltonians that, unlike the transverse Ising model, are not in general believed to be classically efficiently simulatable. Furthermore, they propose new methods for training that allow for parameters corresponding to quantum terms to be learned. Finally, their QBM enables a form of quantum-state tomography that can learn non-classical data, and also generate copies of the reconstructed state. Again, small-scale numerical evidence suggests that their proposals may provide an advantage over classical BMs, although their tendency to overfit data must be investigated further. The practical value of BMs, either classical or quantum, relies in part on the ease with which they are trained. This in turn depends on sampling efficiently from the Boltzmann distribution. Classically this can be done approximately only for the special case of restricted Boltzmann machines (RBM), i.e. where connections are between visible and hidden units, and no connections exist between neurons of the same type. Wiebe et al. [6] propose two quantum algorithms for Boltzmann sampling, based on starting from a state corresponding to the product distribution that is closest to the target Boltzmann distribution. While small-scale numerical studies are encouraging, true bounds on the running time of their algorithms rely on a problem-dependent parameter κ, the scaling of which is hard to estimate. The hope remains though that empirically their algorithms work well in practice, even for multi-layer and fully connected BMs. The expressive power of BMs for compactly representing complicated probability distributions has led to another line of investigation, namely their use for representing quantum many-body wave functions. The authors apply this technique to two prototypical spin models and achieve high accuracy in both ground-state estimation and dynamical evolution, improving on existing state-of-the-art techniques. Their results indicate that RBM states can capture complex many-body correlations in an efficient manner—evidence reinforced by the work of Deng et al. [8], who show that certain topological quantum states in one, two and three dimensions can be represented by RBM states with a number of parameters only linear in the system size. In a subsequent study, however, Gao and Duan [9] prove that although RBM states can indeed represent many highly entangled states, their expressive power is limited in that they cannot represent arbitrary states generated by polynomial-sized quantum circuits, nor those that are the ground states of k-local Hamiltonians with polynomial-sized energy gaps, unless the polynomial hierarchy from computational complexity theory collapses. On the other hand, they show that these two important classes of states can be efficiently represented by a deep Boltzmann machine of three layers. Intriguingly though, another paper by Deng et al. [10] gives an analytic construction of a family of RBM states with a number of parameters linear in the system size that exhibits maximal volume-law entanglement, and that cannot be represented efficiently by either matrix product states (MPS) or tensor-network (TN) states. This shows that the expressive power of even efficiently parametrized RBM states is not limited by entanglement in the same way that other popular compact representations are. This gives the hope that RBM states may prove a valuable tool in understanding entangled quantum systems intractable by traditional methods. Levine et al. [11] establish another connection between the many-body wave function and neural networks, this time in the form of convolutional arithmetic circuits (ConvAC), a variant of the popular convolutional neural networks in widespread use today. By proving a one-to-one equivalence between a ConvAC circuit and TN quantum states, the authors are able to use tools from quantum information theory to analyze the entanglement present in the corresponding TN, and quantify the neural network's ability to model complex correlations between its inputs. This in turn can be used to obtain bounds on the expressivity of a given ConvAC architecture and be of practical use in choosing network parameters tailored to the data. While ConvAC are not widely used in practice, the authors empirically validate that the insights gained can be applied successfully to more common convolutional neural-network designs. Generative adversarial networks (GAN) are machine-learning algorithms in which two players—a generator and a discriminator—compete against each another. The generator's goal is to create and send fake data samples to the discriminator and maximize the probability that the discriminator erroneously believes that the sample comes from some true data distribution. The discriminator attempts to discern real from fake data as best as possible. Under reasonable assumptions the game converges to a point where the discriminator is unable to discriminate with probability greater than 1/2 (i.e. no better than random guessing), and the generator in fact has learned to generate data from the true distribution. Lloyd and Weedbrook [12] consider the natural extension in which one or both of the generator and discriminator are equipped with quantum information processors. When the data to be learned are a quantum state and both parties have quantum processors, the authors show that the unique end point of the adversarial game also coincides with the generator producing a quantum state corresponding to the true data statistics, and the discriminator unable to discern true from false data better than random guessing. They further argue that a different outcome may hold when the true samples are produced by the measurement of some quantum state, and the generator is purely classical. The concept of quantum supremacy is based on the assumption that there exist probability distributions accessible to a quantum computer that no classical computer can efficiently sample from. If the true data correspond to one of these distributions, then no classical generator can efficiently generate fake data reproducing the true statistics and, in principle, the discriminator can successfully discriminate true from fake data. However, whether the discriminator can efficiently find the measurements required to do so is unclear. Quantum-computing techniques have also found applications in autoencoders—neural networks where a larger number of input and output neurons are connected via a smaller number of intermediary latent neurons. The goal is to tune the weights of the network so that a training set of classical bit strings can be approximately recovered after being passed through the autoencoder. Romero et al. [13] consider instead a training set of quantum states, and ask whether a quantum circuit with a similar bottleneck structure can compress and approximately recover the data. The circuits that they consider consist of gates chosen from some set of parametrized unitary operations, and arranged in some fixed order. The parameters are then optimized classically so that the circuit best predicts the training-set labels. The authors demonstrate the validity of this approach by showing that ground states of the Hubbard model and certain molecules can indeed be compressed and recovered by such circuits. However, in general, the performance of such quantum autoencoders will be highly dependent on the kind of programmable circuit on which they are based, and at present there does not seem to be a structured way of determining this. Quantum neural networks are still in their infancy and, while the above works are important steps towards bridging our understanding between classical neural networks and quantum computing, many questions remain unanswered. If the expressivity of Boltzmann machine states is not restricted by entanglement in the same way that MPS and tensor-network states are, what is the best way of understanding their limitations? How can physicists use the correspondence between ConvAC and TN to their advantage in finding efficient, expressive representations of many-body wave functions? Can classical data distributions be learned more efficiently by quantum GAN than classical GAN? Whatever the outcome of these and many other related questions, this promises to be an exciting and active area of research in the years to come.
The astounding acceleration in Artificial Intelligence and Quantum Computing advances naturally gives rise to a line of research, which unrolls the potential advantages of quantum computing on classical Machine Learning tasks, known as Quantum Machine Learning or Quantum Machine Intelligence. The typical objectives are either (1) exploring the potential quantum advantages on classical learning tasks or (2) levering well-established classical ML algorithms to tackle quantum-related problems on Noisy Intermediate-Scale Quantum (NISQ) devices. Along the second research direction, we study Quantum Neural Networks (QNNs) to accomplish the purpose of Bayesian learning. By observing a wide range of studies on QNNs, in which the sole training method is based on frequentist training, we find that Bayesian learning benefits QNNs from two aspects. First, Bayesian-trained models enjoy a high level of generalization due to the prior and posterior distribution usage compared to frequentist training, which will be justified by this paper’s theoretical study of model capacity. Second, Bayesian Inference offers epistemic uncertainty estimation, which merits the decision-making process. It is worth mentioning that frequentist-trained QNNs generally lack this desirable property. Under the Bayesian training procedure, our derived models can be considered a new class of QNNs (called BayesianQNNs) which possesses both desirable properties of Bayesian Inference while maintaining comparable predictive performance as frequentist counterparts. The proposed Bayesian Quantum Neural Networks is justified by empirical evidence from numerical experiments.
One of the main reasons why the physics of quantum many-body systems is hard \nlies in the curse of dimensionality: \nThe number of states of such systems increases exponentially with the number of degrees of freedom involved. \nAs a result, computations for realistic systems become intractable, \nand even numerical methods are limited to comparably small system sizes. \nMany efforts in modern physics research are therefore concerned with \nfinding efficient representations of quantum states and \nclever approximations schemes that would allow them to characterize physical systems of interest. \n \nMeanwhile, Deep Learning (DL) has solved many non-scientific problems \nthat have been unaccessible to conventional methods for a similar reason. \nThe concept underlying DL is to extract knowledge from data \nby identifying patterns and regularities. \nThe remarkable success of DL has excited many physicists about \nthe prospect of leveraging its power to solve intractable problems in physics. \nAt the same time, DL turned out to be an interesting complex many-body \nproblem in itself. \nIn contrast to its widespread empirical applications, \nthe theoretical foundation of DL is strongly underdeveloped. \nIn particular, as long as its decision-making process and result interpretability remain opaque, \nDL can not claim the status of a scientific tool. \n \nIn this thesis, I explore the interface between DL and quantum many-body physics, \nand investigate DL both as a tool and as a subject of study. \nThe first project presented here is a theory-based study of \na fundamental open question about the role of width and the number of parameters in deep neural networks. \nIn this work, we consider a DL setup for the image recognition task on standard benchmarking datasets. \nWe combine controlled experiments with a theoretical analysis, including analytical calculations for a toy model. \nThe other three works focus on the application of Restricted Boltzmann Machines as generative models \nfor the task of wavefunction reconstruction from measurement data on a quantum many-body system. \nFirst, we implement this approach as a software package, making it available as a tool for experimentalists. \nFollowing the idea that physics problems can be used to characterize DL tools, \nwe then use our extensive knowledge of this setup to conduct a systematic study of \nhow the RBM complexity scales with the complexity of the physical system. \nFinally, in a follow-up study we focus on the effects of parameter pruning techniques \non the RBM and its scaling behavior.
Quantum Bayesian Neural Networks (QBNNs) represent a novel approach to neural network architecture, integrating quantum superposition and entanglement to enhance ensemble learning performance, particularly when dealing with high-dimensional data. This paper explores the potential of QBNNs to overcome the limitations of classical ensembles by leveraging quantum phenomena for improved representation, processing, and generalization. We detail the design of a QBNN framework, focusing on the interplay between quantum states, Bayesian inference, and ensemble aggregation. The core claim centers on the design of a QBNN that utilizes quantum superposition and entanglement to achieve significantly enhanced accuracy and robustness in ensemble learning scenarios. The paper presents a comprehensive analysis of the theoretical and practical implications of this approach, including mathematical formulations and experimental validation demonstrating improvements over traditional ensemble methods. We discuss the challenges and future directions for this research, emphasizing the potential for transformative advancements in data analysis and machine learning.