Predicting a stochastic process' future lies at the heart of many scientific areas.A predictive model extracts information from a stochastic process' past and uses it to generate future statistics.There has been significant amount of effort expended towards finding optimal predictive models that minimize the required amount of past information.By taking advantage of quantum resources, quantum models have been shown to reduce the memory requirements beyond classical limits.Meanwhile, tensor networks are an extremely useful mathematical tool for understanding quantum many-body systems.In this thesis, we explore the connection between tensor networks and quantum predictive models.A particular class of tensor networks, matrix product states (MPS), is utilized to further improve quantum models.First, we establish a connection between matrix product states (MPS) and the optimal classical predictive models.MPS methods offer a systematic method for constructing quantum predictive models and even an improved method for computing the amount of quantum memory.Second, we show that for some families of stochastic processes our method allows us to construct quantum models with unbounded memory advantages over the optimal classical models.Third, we propose a family of divergence measures to quantify the distance between two stochastic processes.This family of divergence measures overcomes certain weaknesses of the existing measures.Moreover, we propose an efficient means of computing the divergence measure using our MPS methods.Finally, we address the problem of constructing quantum predictive models solely from the output of an observed stochastic process.We use machine learning methods to propose a systematic algorithm for learning the quantum model.This provides an alternative method for constructing quantum models with fixed amounts of amount of memory.
Predicting a stochastic process' future lies at the heart of many scientific areas.A predictive model extracts information from a stochastic process' past and uses it to generate future statistics.There has been significant amount of effort expended towards finding optimal predictive models that minimize the required amount of past information.By taking advantage of quantum resources, quantum models have been shown to reduce the memory requirements beyond classical limits.Meanwhile, tensor networks are an extremely useful mathematical tool for understanding quantum many-body systems.In this thesis, we explore the connection between tensor networks and quantum predictive models.A particular class of tensor networks, matrix product states (MPS), is utilized to further improve quantum models.First, we establish a connection between matrix product states (MPS) and the optimal classical predictive models.MPS methods offer a systematic method for constructing quantum predictive models and even an improved method for computing the amount of quantum memory.Second, we show that for some families of stochastic processes our method allows us to construct quantum models with unbounded memory advantages over the optimal classical models.Third, we propose a family of divergence measures to quantify the distance between two stochastic processes.This family of divergence measures overcomes certain weaknesses of the existing measures.Moreover, we propose an efficient means of computing the divergence measure using our MPS methods.Finally, we address the problem of constructing quantum predictive models solely from the output of an observed stochastic process.We use machine learning methods to propose a systematic algorithm for learning the quantum model.This provides an alternative method for constructing quantum models with fixed amounts of amount of memory.
In stochastic modeling, there has been a significant effort towards finding predictive models that predict a stochastic process' future using minimal information from its past. Meanwhile, in condensed matter physics, matrix product states (MPS) are known as a particularly efficient representation of 1D spin chains. In this Letter, we associate each stochastic process with a suitable quantum state of a spin chain. We then show that the optimal predictive model for the process leads directly to an MPS representation of the associated quantum state. Conversely, MPS methods offer a systematic construction of the best known quantum predictive models. This connection allows an improved method for computing the quantum memory needed for generating optimal predictions. We prove that this memory coincides with the entanglement of the associated spin chain across the past-future bipartition.
Optimizing hyperparameters in deep learning models remains a significant bottleneck in the development and deployment of effective neural networks. Traditional methods, such as grid search and random search, often struggle with the exponentially growing search space, leading to substantial computational costs and limited exploration of potentially optimal configurations. This paper proposes a novel approach to deep learning hyperparameter optimization that leverages the principles of quantum-inspired optimization, specifically employing tensor network methods, notably Matrix Product States (MPS), to efficiently approximate the complex optimization landscape. Our method utilizes quantum-inspired sampling techniques to guide the exploration, effectively mimicking the probabilistic nature of quantum measurements. We demonstrate the feasibility and potential advantages of this approach through theoretical analysis and a conceptual framework. The core of our method lies in representing the hyperparameter space as a tensor network, allowing for efficient calculations of the objective function (e.g., validation loss) without exhaustively evaluating every possible combination. This significantly reduces the computational burden compared to conventional methods. The paper outlines a detailed algorithm and discusses key considerations for implementation, including tensor network construction, sampling strategies, and convergence criteria. We propose a framework for iteratively refining the tensor network representation based on observed validation performance, ultimately converging towards a high-performing hyperparameter configuration. The theoretical underpinning draws upon concepts from quantum mechanics, particularly the idea of measurement and probability distribution, to drive the optimization process. We believe this quantum-inspired approach offers a promising avenue for accelerating deep learning hyperparameter tuning and achieving state-of-the-art performance.
The Thesis explores the application of Matrix Product States (MPS) in the domain of quantum many-body systems and quantum computing, show casing the versatility and effectiveness of MPS for addressing complex com putational challenges. The opening chapter introduces essential tools for Tensor Networks, addressing the curse of dimensionality and elucidating the main characteristics of Matrix Product States and Matrix Product Op erators. The discussion encompasses also important numerical techniques, including Density Matrix Renormalization Group and Time Evolving Block Decimation. Furthermore, the relation between MPS and quantum circuits is discussed. The subsequent Chapters delve into specific applications of MPS. In the second Chapter, we introduce a Tensor Network ansatz inspired by the backflow transformation for correlated systems. This extension of the MPS representation ensures an area law for entanglement in dimensions one or greater. We employ an optimization scheme combining DMRG and variational Monte Carlo algorithms for efficient ground-state search. Bench marking against spin models demonstrates high accuracy. The third Chapter explores quantum annealing for optimizing complex classical spin Hamil tonians, as the Hopfield model and the binary perceptron. We introduce an efficient Tensor Network representation for the adiabatic time evolution of quantum annealing, enabling scalable classical simulations. The use of MPS in mitigating Trotter errors and mapping to quantum circuits is also explored. In the fourth Chapter we present a novel method for evaluating the amount of nonstabilizerness (also known as quantum magic) contained in an MPS. We overcome the exponentially hard evaluation of this quan tity by employing a simple perfect sampling technique of Pauli string. Our innovative MPS approach enables efficient computation. Benchmarked on magic states and the quantum Ising chain ground state, this method offers easy access to the non-equilibrium dynamics of nonstabilizerness following a quantum quench. The Appendices provide additional insights, including: an introduction to variational Monte Carlo, technical details on MPS simula tions for the quantum annealing and a comprehensive overview on stabilizer formalism and nonstabilizerness.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
Modeling joint probability distributions over sequences has been studied from many perspectives. The physics community developed matrix product states, a tensor-train decomposition for probabilistic modeling, motivated by the need to tractably model many-body systems. But similar models have also been studied in the stochastic processes and weighted automata literature, with little work on how these bodies of work relate to each other. We address this gap by showing how stationary or uniform versions of popular quantum tensor network models have equivalent representations in the stochastic processes and weighted automata literature, in the limit of infinitely long sequences. We demonstrate several equivalence results between models used in these three communities: (i) uniform variants of matrix product states, Born machines and locally purified states from the quantum tensor networks literature, (ii) predictive state representations, hidden Markov models, norm-observable operator models and hidden quantum Markov models from the stochastic process literature,and (iii) stochastic weighted automata, probabilistic automata and quadratic automata from the formal languages literature. Such connections may open the door for results and methods developed in one area to be applied in another.
We show how a tensor-network-based machine learning algorithm can learn the structures of generic, non-Markovian, quantum stochastic processes. First, a process is represented as a matrix product operator (MPO) and trained with a database of local input states at different times and the corresponding time-nonlocal output state. We then apply the algorithm to predict the output state of a process at different times for a system that is coupled to a spin-chain environment. We then reconstruct the full process, and we quantify the non-Markovian memory by means of the bond dimension of the MPO for various properties of the system, of the environment, and of their interaction. Our study paves the way for a possible experimental investigation into the process tensor and its properties, and an effective characterization of noise in quantum devices.