Squared tensor networks (TNs) and their generalization as parameterized computational graphs -- squared circuits -- have been recently used as expressive distribution estimators in high dimensions. However, the squaring operation introduces additional complexity when marginalizing variables or computing the partition function, which hinders their usage in machine learning applications. Canonical forms of popular TNs are parameterized via unitary matrices as to simplify the computation of particular marginals, but cannot be mapped to general circuits since these might not correspond to a known TN. Inspired by TN canonical forms, we show how to parameterize squared circuits to ensure they encode already normalized distributions. We then use this parameterization to devise an algorithm to compute any marginal of squared circuits that is more efficient than a previously known one. We conclude by formally showing the proposed parameterization comes with no expressiveness loss for many circuit classes.
Modern approaches to generative modeling of continuous data using tensor networks incorporate compression layers to capture the most meaningful features of high-dimensional inputs. These methods, however, rely on traditional Matrix Product States (MPS) architectures. Here, we demonstrate that beyond a certain threshold in data and bond dimensions, a comb-shaped tensor network architecture can yield more efficient contractions than a standard MPS. This finding suggests that for continuous and high-dimensional data distributions, transitioning from MPS to a comb tensor network representation can substantially reduce computational overhead while maintaining accuracy.
Logic Tensor Networks (LTN) is a Neuro-Symbolic framework that effectively incorporates deep learning and logical reasoning. In particular, LTN allows defining a logical knowledge base and using it as the objective of a neural model. This makes learning by logical reasoning possible as the parameters of the model are optimized by minimizing a loss function composed of a set of logical formulas expressing facts about the learning task. The framework learns via gradient-descent optimization. Fuzzy logic, a relaxation of classical logic permitting continuous truth values in the interval [0,1], makes this learning possible. Specifically, the training of an LTN consists of three steps. Firstly, (1) the training data is used to ground the formulas. Then, (2) the formulas are evaluated, and the loss function is computed. Lastly, (3) the gradients are back-propagated through the logical computational graph, and the weights of the neural model are changed so the knowledge base is maximally satisfied. LTNtorch is the fully documented and tested PyTorch implementation of Logic Tensor Networks. This paper presents the formalization of LTN and how LTNtorch implements it. Moreover, it provides a basic binary classification example.
Tensor Networks (TNs) are a computational paradigm used for representing quantum many-body systems. Recent works have shown how TNs can also be applied to perform Machine Learning (ML) tasks, yielding comparable results to standard supervised learning techniques. In this work, we study the use of Tree Tensor Networks (TTNs) in high-frequency real-time applications by exploiting the low-latency hardware of the Field-Programmable Gate Array (FPGA) technology. We present different implementations of TTN classifiers, capable of performing inference on classical ML datasets as well as on complex physics data. A preparatory analysis of bond dimensions and weight quantization is realized in the training phase, together with entanglement entropy and correlation measurements, that help setting the choice of the TTN architecture. The generated TTNs are then deployed on a hardware accelerator; using an FPGA integrated into a server, the inference of the TTN is completely offloaded. Eventually, a classifier for High Energy Physics (HEP) applications is implemented and executed fully pipelined with sub-microsecond latency.
We present a series of algorithms in tensor networks for anomaly detection in datasets, by using data compression in a Tensor Train representation. These algorithms consist of preserving the structure of normal data in compression and deleting the structure of anomalous data. The algorithms can be applied to any tensor network representation. We test the effectiveness of the methods with digits and Olivetti faces datasets and a cybersecurity dataset to determine cyber-attacks.
We study the identification of quantum phases of matter, at zero temperature, when only part of the phase diagram is known in advance. Following a supervised learning approach, we show how to use our previous knowledge to construct an observable capable of classifying the phase even in the unknown region. By using a combination of classical and quantum techniques, such as tensor networks, kernel methods, generalization bounds, quantum algorithms, and shadow estimators, we show that, in some cases, the certification of new ground states can be obtained with a polynomial number of measurements. An important application of our findings is the classification of the phases of matter obtained in quantum simulators, e.g., cold atom experiments, capable of efficiently preparing ground states of complex many-particle systems and applying simple measurements, e.g., single qubit measurements, but unable to perform a universal set of gates.
Fourier Neural Operators (FNOs) excel on tasks using functional data, such as those originating from partial differential equations. Such characteristics render them an effective approach for simulating the time evolution of quantum wavefunctions, which is a computationally challenging, yet coveted task for understanding quantum systems. In this manuscript, we use FNOs to model the evolution of random quantum spin systems, so chosen due to their representative quantum dynamics and minimal symmetry. We explore two distinct FNO architectures and examine their performance for learning and predicting time evolution using both random and low-energy input states. Additionally, we apply FNOs to a compact set of Hamiltonian observables ($\sim\text{poly}(n)$) instead of the entire $2^n$ quantum wavefunction, which greatly reduces the size of our inputs and outputs and, consequently, the requisite dimensions of the resulting FNOs. Moreover, this Hamiltonian observable-based method demonstrates that FNOs can effectively distill information from high-dimensional spaces into lower-dimensional spaces. The extrapolation of Hamiltonian observables to times later than those used in training is of particular interest, as this stands to fundamentally increase the simulatability of quantum systems past both the coherence times of contemporary quantum architectures and the circuit-depths of tractable tensor networks.
The growing computational demands of artificial intelligence (AI) in addressing climate change raise significant concerns about inefficiencies and environmental impact, as highlighted by the Jevons paradox. We propose an attention-enhanced quantum physics-informed neural networks model (AQ-PINNs) to tackle these challenges. This approach integrates quantum computing techniques into physics-informed neural networks (PINNs) for climate modeling, aiming to enhance predictive accuracy in fluid dynamics governed by the Navier-Stokes equations while reducing the computational burden and carbon footprint. By harnessing variational quantum multi-head self-attention mechanisms, our AQ-PINNs achieve a 51.51% reduction in model parameters compared to classical multi-head self-attention methods while maintaining comparable convergence and loss. It also employs quantum tensor networks to enhance representational capacity, which can lead to more efficient gradient computations and reduced susceptibility to barren plateaus. Our AQ-PINNs represent a crucial step towards more sustainable and effective climate modeling solutions.
Out-of-Time Ordered Correlators (OTOCs) are widely used to investigate information scrambling in quantum systems. However, directly computing OTOCs with classical computers is an expensive procedure. This is due to the need to classically simulate the dynamics of quantum many-body systems, which entails computational costs that scale rapidly with system size. Similarly, exact simulation of the dynamics with a quantum computer (QC) will either only be possible for short times with noisy intermediate-scale quantum (NISQ) devices, or will require a fault-tolerant QC which is currently beyond technological capabilities. This motivates a search for alternative approaches to determine OTOCs and related quantities. In this study, we explore four parameterised sets of Hamiltonians describing local one-dimensional quantum systems of interest in condensed matter physics. For each set, we investigate whether classical kernel methods (KMs) can accurately learn the XZ-OTOC and a particular sum of OTOCs, as functions of the Hamiltonian parameters. We frame the problem as a regression task, generating small batches of labelled data with classical tensor network methods for quantum many-body systems with up to 40 qubits. Using this data, we train a variety of standard kernel machines and observe that the Laplacian and radial basis function (RBF) kernels perform best, achieving a coefficient of determination (\(R^2\)) on the testing sets of at least 0.7167, with averages between 0.8112 and 0.9822 for the various sets of Hamiltonians, together with small root mean squared error and mean absolute error. Hence, after training, the models can replace further uses of tensor networks for calculating an OTOC function of a system within the parameterised sets. Accordingly, the proposed method can assist with extensive evaluations of an OTOC function.
Quantum many-body systems are of great interest for many research areas, including physics, biology and chemistry. However, their simulation is extremely challenging, due to the exponential growth of the Hilbert space with the system size, making it exceedingly difficult to parameterize the wave functions of large systems by using exact methods. Neural networks and machine learning in general are a way to face this challenge. For instance, methods like Tensor networks and Neural Quantum States are being investigated as promising tools to obtain the wave function of a quantum mechanical system. In this tutorial, we focus on a particularly promising class of deep learning algorithms. We explain how to construct a Physics-Informed Neural Network (PINN) able to solve the Schr\"odinger equation for a given potential, by finding its eigenvalues and eigenfunctions. This technique is unsupervised, and utilizes a novel computational method in a manner that is barely explored. PINNs are a deep learning method that exploits Automatic Differentiation to solve Integro-Differential Equations in a mesh-free way. We show how to find both the ground and the excited states. The method discovers the states progressively by starting from the ground state. We explain how to introduce inductive biases in the loss to exploit further knowledge of the physical system. Such additional constraints allow for a faster and more accurate convergence. This technique can then be enhanced by a smart choice of collocation points in order to take advantage of the mesh-free nature of the PINN. The methods are made explicit by applying them to the infinite potential well and the particle in a ring, a challenging problem to be learned by an Artificial Intelligence agent due to the presence of complex-valued eigenfunctions and degenerate states.
Quantum machine learning (QML) is a rapidly expanding field that merges the principles of quantum computing with the techniques of machine learning. One of the powerful mathematical frameworks in this domain is tensor networks. These networks are used to approximate high-order tensors by contracting tensors with lower ranks. Initially developed for simulating quantum systems, tensor networks have become integral to quantum computing and, by extension, to QML. Drawing inspiration from these quantum methods, specifically the Matrix Product States (MPS), we apply them in a classical machine learning setting. Their ability to efficiently represent and manipulate complex, high-dimensional data makes them effective in a supervised learning framework. Here, we present an MPS model, in which the MPS functions as both a classifier and a generator. The dual functionality of this novel MPS model permits a strategy that enhances the traditional training of supervised MPS models. This framework is inspired by generative adversarial networks and is geared towards generating more realistic samples by reducing outliers. In addition, our contributions offer insights into the mechanics of tensor network methods for generation tasks. Specifically, we discuss alternative embedding functions and a new sampling method from non-normalized MPSs.
In this paper, a deep neural network approach and a neuro-symbolic one are proposed for classification and regression. The neuro-symbolic predictive models based on Logic Tensor Networks are capable of discriminating and in the same time of explaining the characterization of bad connections, called alerts or attacks, and of normal connections. The proposed hybrid systems incorporate both the ability of deep neural networks to improve on their own through experience and the interpretability of the results provided by symbolic artificial intelligence approach. To justify the need for shifting towards hybrid systems, explanation, implementation, and comparison of the dense neural network and the neuro-symbolic network is performed in detail. For the comparison to be relevant, the same datasets were used in training and the metrics resulted have been compared. A review of the resulted metrics shows that while both methods have similar precision in their predictive models, with Logic Tensor Networks being also possible to have interactive accuracy and deductive reasoning over data. Other advantages and disadvantages such as overfitting mitigation and scalability issues are also further discussed.
The ability to learn compact, high-quality, and easy-to-optimize representations for visual data is paramount to many applications such as novel view synthesis and 3D reconstruction. Recent work has shown substantial success in using tensor networks to design such compact and high-quality representations. However, the ability to optimize tensor-based representations, and in particular, the highly compact tensor train representation, is still lacking. This has prevented practitioners from deploying the full potential of tensor networks for visual data. To this end, we propose 'Prolongation Upsampling Tensor Train (PuTT)', a novel method for learning tensor train representations in a coarse-to-fine manner. Our method involves the prolonging or `upsampling' of a learned tensor train representation, creating a sequence of 'coarse-to-fine' tensor trains that are incrementally refined. We evaluate our representation along three axes: (1). compression, (2). denoising capability, and (3). image completion capability. To assess these axes, we consider the tasks of image fitting, 3D fitting, and novel view synthesis, where our method shows an improved performance compared to state-of-the-art tensor-based methods. For full results see our project webpage: https://sebulo.github.io/PuTT_website/
Modern quantum optical systems such as photonic quantum computers and quantum imaging devices require great precision in their designs and implementations in the hope to realistically exploit entanglement and reach a real quantum advantage. The theoretical and experimental explorations and validations of these systems are greatly dependent on the precision of our classical simulations. However, as Hilbert spaces increases, traditional computational methods used to design and optimize these systems encounter hard limitations due to the quantum curse of dimensionally. To address this challenge, we propose an approach based on neural and tensor networks to approximate the exact unitary evolution of closed entangled systems in a precise, efficient and quantum consistent manner. By training the networks with a reasonably small number of examples of quantum dynamics, we enable efficient parameter estimation in larger Hilbert spaces, offering an interesting solution for a great deal of quantum metrology problems.
Learning faithful representations of quantum states is crucial to fully characterizing the variety of many-body states created on quantum processors. While various tomographic methods such as classical shadow and MPS tomography have shown promise in characterizing a wide class of quantum states, they face unique limitations in detecting topologically ordered two-dimensional states. To address this problem, we implement and study a heuristic tomographic method that combines variational optimization on tensor networks with randomized measurement techniques. Using this approach, we demonstrate its ability to learn the ground state of the surface code Hamiltonian as well as an experimentally realizable quantum spin liquid state. In particular, we perform numerical experiments using MPS ans\"atze and systematically investigate the sample complexity required to achieve high fidelities for systems of sizes up to $48$ qubits. In addition, we provide theoretical insights into the scaling of our learning algorithm by analyzing the statistical properties of maximum likelihood estimation. Notably, our method is sample-efficient and experimentally friendly, only requiring snapshots of the quantum state measured randomly in the $X$ or $Z$ bases. Using this subset of measurements, our approach can effectively learn any real pure states represented by tensor networks, and we rigorously prove that random-$XZ$ measurements are tomographically complete for such states.
Probabilistic inference is a fundamental task in modern machine learning. Recent advances in tensor network (TN) contraction algorithms have enabled the development of better exact inference methods. However, many common inference tasks in probabilistic graphical models (PGMs) still lack corresponding TN-based adaptations. In this work, we advance the connection between PGMs and TNs by formulating and implementing tensor-based solutions for the following inference tasks: (i) computing the partition function, (ii) computing the marginal probability of sets of variables in the model, (iii) determining the most likely assignment to a set of variables, and (iv) the same as (iii) but after having marginalized a different set of variables. We also present a generalized method for generating samples from a learned probability distribution. Our work is motivated by recent technical advances in the fields of quantum circuit simulation, quantum many-body physics, and statistical physics. Through an experimental evaluation, we demonstrate that the integration of these quantum technologies with a series of algorithms introduced in this study significantly improves the effectiveness of existing methods for solving probabilistic inference tasks.
In this study, we introduce a novel family of tensor networks, termed constrained matrix product states (MPS), designed to incorporate exactly arbitrary discrete linear constraints, including inequalities, into sparse block structures. These tensor networks are particularly tailored for modeling distributions with support strictly over the feasible space, offering benefits such as reducing the search space in optimization problems, alleviating overfitting, improving training efficiency, and decreasing model size. Central to our approach is the concept of a quantum region, an extension of quantum numbers traditionally used in U(1) symmetric tensor networks, adapted to capture any linear constraint, including the unconstrained scenario. We further develop a novel canonical form for these new MPS, which allow for the merging and factorization of tensor blocks according to quantum region fusion rules and permit optimal truncation schemes. Utilizing this canonical form, we apply an unsupervised training strategy to optimize arbitrary objective functions subject to discrete linear constraints. Our method's efficacy is demonstrated by solving the quadratic knapsack problem, achieving superior performance compared to a leading nonlinear integer programming solver. Additionally, we analyze the complexity and scalability of our approach, demonstrating its potential in addressing complex constrained combinatorial optimization problems.
Healthcare industries frequently handle sensitive and proprietary data, and due to strict privacy regulations, they are often reluctant to share data directly. In today's context, Federated Learning (FL) stands out as a crucial remedy, facilitating the rapid advancement of distributed machine learning while effectively managing critical concerns regarding data privacy and governance. The fusion of federated learning and quantum computing represents a groundbreaking interdisciplinary approach with immense potential to revolutionize various industries, from healthcare to finance. In this work, we proposed a federated learning framework based on quantum tensor networks, which leverages the principles of many-body quantum physics. Currently, there are no known classical tensor networks implemented in federated settings. Furthermore, we investigated the effectiveness and feasibility of the proposed framework by conducting a differential privacy analysis to ensure the security of sensitive data across healthcare institutions. Experiments on popular medical image datasets show that the federated quantum tensor network model achieved a mean receiver-operator characteristic area under the curve (ROC-AUC) between 0.91-0.98. Experimental results demonstrate that the quantum federated global model, consisting of highly entangled tensor network structures, showed better generalization and robustness and achieved higher testing accuracy, surpassing the performance of locally trained clients under unbalanced data distributions among healthcare institutions.
In this paper we present a study of the applicability and feasibility of quantum-inspired algorithms and techniques in tensor networks for industrial environments and contexts, with a compilation of the available literature and an analysis of the use cases that may be affected by such methods. In addition, we explore the limitations of such techniques in order to determine their potential scalability.
A long-standing issue in mathematical finance is the speed-up of option pricing, especially for multi-asset options. A recent study has proposed to use tensor train learning algorithms to speed up Fourier transform (FT)-based option pricing, utilizing the ability of tensor trains to compress high-dimensional tensors. Another usage of the tensor train is to compress functions, including their parameter dependence. Here, we propose a pricing method, where, by a tensor train learning algorithm, we build tensor trains that approximate functions appearing in FT-based option pricing with their parameter dependence and efficiently calculate the option price for the varying input parameters. As a benchmark test, we run the proposed method to price a multi-asset option for the various values of volatilities and present asset prices. We show that, in the tested cases involving up to 11 assets, the proposed method outperforms Monte Carlo-based option pricing with $10^6$ paths in terms of computational complexity while keeping better accuracy.