Quantum systems are promising candidates of future computing and information processing devices. In a large system, information about the quantum states and processes may be incomplete and scattered. To integrate the distributed information we propose a quantum version of probabilistic graphical models. Variables in the model (quantum states and measurement outcomes) are linked by several types of operators (unitary, measurement, and merge/split operators). We propose algorithms for three machine learning tasks in quantum probabilistic graphical models: a belief propagation algorithm for inference of unknown states, an iterative algorithm for simultaneous estimation of parameter values and hidden states, and an active learning algorithm to select measurement operators based on observed evidence. We validate these algorithms on simulated data and point out future extensions toward a more comprehensive theory of quantum probabilistic graphical models.
Quantum systems are promising candidates of future computing and information processing devices. In a large system, information about the quantum states and processes may be incomplete and scattered. To integrate the distributed information we propose a quantum version of probabilistic graphical models. Variables in the model (quantum states and measurement outcomes) are linked by several types of operators (unitary, measurement, and merge/split operators). We propose algorithms for three machine learning tasks in quantum probabilistic graphical models: a belief propagation algorithm for inference of unknown states, an iterative algorithm for simultaneous estimation of parameter values and hidden states, and an active learning algorithm to select measurement operators based on observed evidence. We validate these algorithms on simulated data and point out future extensions toward a more comprehensive theory of quantum probabilistic graphical models.
Belief Propagation algorithms acting on Graphical Models of classical probability distributions, such as Markov Networks, Factor Graphs and Bayesian Networks, are amongst the most powerful known methods for deriving probabilistic inferences amongst large numbers of random variables. This paper presents a generalization of these concepts and methods to the quantum case, based on the idea that quantum theory can be thought of as a noncommutative, operator-valued, generalization of classical probability theory. Some novel characterizations of quantum conditional independence are derived, and definitions of Quantum n-Bifactor Networks, Markov Networks, Factor Graphs and Bayesian Networks are proposed. The structure of Quantum Markov Networks is investigated and some partial characterization results are obtained, along the lines of the Hammersely-Clifford theorem. A Quantum Belief Propagation algorithm is presented and is shown to converge on 1-Bifactor Networks and Markov Networks when the underlying graph is a tree. The use of Quantum Belief Propagation as a heuristic algorithm in cases where it is not known to converge is discussed. Applications to decoding quantum error correcting codes and to the simulation of many-body quantum systems are described.
This paper explores the potential of quantum machine learning (QML) for efficient and accurate probabilistic model learning. Traditional machine learning algorithms often struggle with computational complexity and scalability, particularly when dealing with high-dimensional data and complex probability distributions. This research proposes a novel approach leveraging principles from quantum information theory to design a QML algorithm capable of learning probabilistic models. The core mechanism involves representing model parameters as quantum states and utilizing quantum measurements and entanglement to perform parameter estimation. We aim to demonstrate that this quantum-based method can overcome limitations inherent in classical approaches, potentially leading to significant improvements in both computational efficiency and predictive accuracy. This work presents a foundational framework for further research and development in this emerging field.
This paper explores the application of probabilistic graphical models for dynamic quantum state decoding, aiming to enhance the reliability of quantum information transmission. The core concept involves constructing a quantum state decoder based on a probabilistic graphical model, leveraging measurement results to infer the original quantum state. A Bayesian optimization algorithm is then employed to refine the model's parameters, adapting to evolving measurement data and improving decoding accuracy. The proposed approach offers a novel framework for understanding and optimizing the inherently probabilistic nature of quantum state decoding. The key contributions lie in the integration of graphical models with Bayesian optimization, providing a flexible and adaptive solution compared to traditional decoding methods. This work demonstrates the potential for improved quantum communication protocols by dynamically adjusting the decoding process based on observed data.
Among several tasks in Machine Learning, is the problem of inferring the latent variables of a system and their causal relations with the observed behavior. A paradigmatic instance of such problem is the task of inferring the Hidden Markov Model underlying a given stochastic process. This is known as the positive realization problem (PRP) [Benvenuti,Farina(2004)] and constitutes a central problem in machine learning. The PRP and its solutions have far-reaching consequences in many areas of systems and control theory, and is nowadays an important piece in the broad field of positive systems theory [Luenberger(1979)]. We consider the scenario where the latent variables are quantum (e.g., quantum states of a finite-dimensional system), and the system dynamics is constrained only by physical transformations on the quantum system. The observable dynamics is then described by a quantum instrument, and the task is to determine which quantum instrument-if any-yields the process at hand by iterative application. We take as a starting point the theory of quasi-realizations, whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilities are given by linear functionals on the state vectors. This description, despite its remarkable resemblance with the Hidden Markov Model, or the iterated quantum instrument, is however devoid from any stochastic or quantum mechanical interpretation, as said maps fail to satisfy any positivity conditions. The Completely-Positive realization problem then consists in determining whether an equivalent quantum mechanical description of the same process exists. We generalize some key results of stochastic realization theory, and show that the problem has deep connections with operator systems theory, giving possible insight to the lifting problem in quotient operator systems. Our results have potential applications in quantum machine learning, device-independent characterization and reverse-engineering of stochastic processes and quantum processors, and more generally, of dynamical processes with quantum memory [Guta(2011), Guta&Yamamoto(2013)].
This project introduces a computational framework for quantum trajectory reconstruction using weak measurements and probabilistic inference. In standard quantum mechanics, physical systems are described by a wavefunction, and observable quantities are inherently probabilistic. As a result, particle trajectories are not directly accessible. This work proposes an operational approach in which trajectories are treated as latent variables inferred from noisy measurement data. The framework combines three core components: A numerical solver for the time-dependent Schrödinger equation (using FFT-based split-operator methods) A weak measurement model that simulates continuous, noisy observations of the system Inference engines, including Sequential Monte Carlo (particle filtering) and recurrent neural networks, to reconstruct the most probable trajectory The central idea is to reformulate quantum dynamics as an inference problem over hidden states, where the goal is to estimate the trajectory that best explains the observed measurements. This approach does not violate the principles of quantum mechanics, including the Principe d'incertitude de Heisenberg, but instead provides a new computational lens through which quantum systems can be analyzed. Potential applications include: Quantum sensing and metrology Quantum control and error correction Hybrid AI-quantum systems Advanced simulation of open quantum systems The project is fully reproducible and open-source, with modular architecture designed for research extensibility.
Abstract Probabilistic graphical models play a crucial role in machine learning and have wide applications across various fields. A pivotal subset within this realm is undirected graphical models, also known as Markov random fields. In this work, we explore the application of quantum computing to the field of machine learning, specifically focusing on the task of structure learning in Markov random fields. We propose a quantum algorithm for structure learning of an r-wise Markov Random Field with a bounded degree underlying graph, based on a nearly optimal classical greedy algorithm. Our quantum algorithm provides a polynomial speed-up over the classical counterpart, with a runtime of O ˜ 2 L M n r + 1 log ( 1 / w ) compared to the classical O ˜ M n r , where n is the number of variables. It is nearly a quadratic speedup in terms of n. Through this research, we aim to showcase the potential advantages of quantum computation in tackling complex machine-learning problems. Furthermore, the sample complexity O ˜ 2 2 L τ 2 δ 2 L ( log ( 1 / w ) + log n ) remains consistent between classical and quantum approaches, ensuring the efficiency and robustness of our method. By demonstrating the efficiency and scalability of our quantum algorithm for structure learning, we contribute to the growing body of work that highlights the role of quantum computing in advancing the field of machine learning and probabilistic graphical models.