We derive machine learning algorithms from discretized Euclidean field theories, making inference and learning possible within dynamics described by quantum field theory. Specifically, we demonstrate that the ${\ensuremath{\phi}}^{4}$ scalar field theory satisfies the Hammersley-Clifford theorem, therefore recasting it as a machine learning algorithm within the mathematically rigorous framework of Markov random fields. We illustrate the concepts by minimizing an asymmetric distance between the probability distribution of the ${\ensuremath{\phi}}^{4}$ theory and that of target distributions, by quantifying the overlap of statistical ensembles between probability distributions and through reweighting to complex-valued actions with longer-range interactions. Neural network architectures are additionally derived from the ${\ensuremath{\phi}}^{4}$ theory which can be viewed as generalizations of conventional neural networks and applications are presented. We conclude by discussing how the proposal opens up a new research avenue, that of developing a mathematical and computational framework of machine learning within quantum field theory.
We derive machine learning algorithms from discretized Euclidean field theories, making inference and learning possible within dynamics described by quantum field theory. Specifically, we demonstrate that the \ensuremathϕ4 scalar field theory satisfies the Hammersley-Clifford theorem, therefore recasting it as a machine learning algorithm within the mathematically rigorous framework of Markov random fields. We illustrate the concepts by minimizing an asymmetric distance between the probability distribution of the \ensuremathϕ4 theory and that of target distributions, by quantifying the overlap of statistical ensembles between probability distributions and through reweighting to complex-valued actions with longer-range interactions. Neural network architectures are additionally derived from the \ensuremathϕ4 theory which can be viewed as generalizations of conventional neural networks and applications are presented. We conclude by discussing how the proposal opens up a new research avenue, that of developing a mathematical and computational framework of machine learning within quantum field theory.
Within the past decade, machine learning algorithms have been proposed as a po-tential solution to a variety of research problems which emerge within physics. As this cross-fertilization matures, one is able to investigate if the efficiency of machine learning algorithms can be increased by interpreting them physically and if there exist fundamental connections that can be established between the two research fields. In this thesis, we pursue research directions intimately related to the above questions.First, we investigate the practical implications of interpreting machine learning functions as statistical-mechanical observables. Through this perspective, we explore if we can extend the classification capabilities of machine learning algorithms and if we are able to include neural networks within Hamiltonians to induce phase transitions in systems. A related direction concerns the use of machine learning to construct inverse renormalization group transformations to arbitrarily increase the size of a system. These techniques are then utilized to study the infinite volume limit of discrete spin systems and of quantum field theories, in order to investigate if machine learning is a powerful tool to study phase transitions.In another research direction we explore the derivation of machine learning algo-rithms from quantum field theories. We investigate if the φ4 scalar field theory satisfies the Hammersley-Clifford theorem and if it can be recast as a Markov random field. We then explore if φ4 neural networks can be derived that generalize a certain class of standard neural network architectures, and we present relevant numerical appli-cations. Finally, we discuss how this research direction opens up the opportunity to investigate machine learning within quantum field theory and how it solidifies a rig-orous connection between the research fields of machine learning, probability theory, statistical mechanics, lattice and constructive quantum field theory.
The intersection of machine learning and quantum computing has been one of the greatest scientific projects in current time. With quantum computing in the offing, the assurance of using quantum mechanical effects in computational learning to transform the world has never been closer. The publication of this book comes at a critical time, when theoretical models are shouldering maturity with capabilities of experimentation, that has brought about great possibilities to implement computational benefits that quantum systems can provide in performing machine learning tasks. Quantum machine learning is one field which has rapidly changed in the last ten years. What was originally viewed as mere theoretical conjecture has now evolved into a living research community which includes rigorous mathematical modeling, novel algorithm design, and more and more complex experimental realisations on the near-term quantum computers. Such change can be characterised by the increasing sophistication of our theory in as well as the revolutionary advances in quantum computing technology, and by the noisy intermediate-scale quantum (NISQ) era. Chapters in this book chronologically discuss the complex nature of quantum machine learning, including theoretical principles and challenges of practice. All the chapters cover a very important aspect in this growing area leaving the reader not only with an in-depth analysis but also a wide enough point of view to traverse this somewhat complicated inter-disciplinary area. The book is organized in such a way that that it takes the reader through more advanced parts of the quantum machine learning theory and practice. We start by explaining quantum feature maps, and kernel-type methods and laying the mathematical background on which quantum-enhanced learning is based. There we discuss computational complexity, trainability issues, neural network architectures, scaling of resources, sample complexity limits and the issue critical of quantum data encoding. This book has a variety of audience. The works will also offer exhaustive proposals of key concepts, methodologies and open problems to graduate students and researchers venturing into the field. Individuals who operate quantum computing platforms will have an understanding of the options available when it comes to designing an algorithm and the resources needed, as well as the expectations of its performance. The systematic approach of tackling complexity theorist-foundations and rigorous mathematical analyses will be appreciated by the theorists. Quantum machine learning is interdisciplinary and requires the services and knowledge of quantum physics, computer science, mathematics, and statistics. Although we do not expect our readers to have the basic knowledge of quantum computing and classical machine learning, we give adequate background and references to accommodate other readers of varying technical background. By 2026, quantum machine learning will be on a important crossroads. NISQ has also provided its own stunning demonstrations and harsh lessons on the difficulty of obtaining useful quantum advantage. Other phenomena like barren plateaus have changed the minds of us on the nature of variational quantum algorithm trainability. Resource-efficient quantum learning has been transformed by the advanced measurement protocols, such as classical shadow tomography. The frontier of the implementable quantum algorithms is constantly being pushed by hardware advances. The book reflects the picture of the field in this dynamic time, including the latest theoretical developments, their experiments, and new optimal practices. We admit that quantum machine learning is a fast-developing science, and there are certain spheres of our knowledge that will surely become more profound in the nearest future. The transformative potential of quantum machine learning is still achievable only with further achievements in many dimensions: the theory of when and why quantum algorithms are effective, algorithmic innovations to overcome the limitations of near-term hardware, hardware engineering to overcome qubit quality and system scale, and discovering applications when quantum algorithms and methods and methods offer significant superior practical performance. Hopefully, this book will provide useful guidance to all the existing knowledge and guide to the future innovations. The opportunities are out of the world, yet the challenges are significant. With the ongoing development of quantum hardware capabilities and an ever-increasing theoretical basis, it is quite possible that quantum machine learning will transform the world of approaching the problem of complex computational learning.
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.
This paper investigates the application of quantum entanglement to probability field theory, proposing a novel computational framework predicated on the principles of quantum superposition and entanglement. We explore the potential of this approach to enhance the efficiency of probabilistic computation, specifically focusing on its implications for machine learning. The core of this work centers on the conceptualization of probability events as quantum states, leveraging entanglement to facilitate complex computations and ultimately, explore its applicability to machine learning tasks. The paper details the underlying mathematical foundations and potential benefits of this new computational paradigm, offering a significant departure from traditional probability modeling approaches.
Machine learning improved by quantum computing. Machine learning and quantum physics fix AI and computers. This chapter discusses quantum machine learning theory, methods, and applications. Part 1 thoroughly discusses quantum and classical machine learning. The authors demonstrate how quantum supports vector machines, neural networks, and clustering speed AI. The chapter examines quantum machine learning's merits and downsides. Quantum computers optimize, parallelize, and manage huge data better. Quantum hardware restrictions and error correction reduce noise and decoherence. Explore quantum machine learning in NLP, drug discovery, financial modeling, and image recognition. Many fields could change quantum platform machine learning models with quantum algorithms. The chapter concludes with quantum machine learning directions and challenges. Check trustworthy quantum machine learning frameworks, benchmarks, and hybrid algorithms. Hot: quantum machine learning. This chapter covers fundamentals, research frameworks, and applications.
In this thesis, we investigate whether quantum algorithms can be used in the field of machine learning for both long and near term quantum computers. We will first recall the fundamentals of machine learning and quantum computing and then describe more precisely how to link them through linear algebra: we introduce quantum algorithms to efficiently solve tasks such as matrix product or distance estimation. These results are then used to develop new quantum algorithms for unsupervised machine learning, such as k-means and spectral clustering. This allows us to define many fundamental procedures, in particular in vector and graph analysis. We will also present new quantum algorithms for neural networks, or deep learning. For this, we introduce an algorithm to perform a quantum convolution product on images, as well as a new way to perform a fast tomography on quantum states. We prove that these quantum algorithms are faster versions of equivalent classical algorithms, but exhibit random effects due to the quantum nature of the computation. Many simulations have been carried out to study these effects and measure their learning accuracy on real data. Finally, we will present a quantum orthogonal neural network circuit adapted to the currently available small and imperfect quantum computers. This allows us to perform real experiments to test our theory.