Graphical models of probabilistic dependencies have been extensively investigated in the context of classical uncer-tainty. However, in some domains (most notably, in compu-tational physics and quantum computing) the nature of the relevant uncertainty is non-classical, and the laws of clas-sical probability theory are superseded by those of quan-tum mechanics. In this paper we introduce Markov Entan-glement Networks (MEN), a novel class of graphical repre-sentations of quantum-mechanical dependencies in the con-text of such non-classical systems. MEN are the quantum-mechanical analogue of Markovian Networks, a family of undirected graphical representations which, in the classical domain, exploit a notion of conditional independence among subsystems. After defining a notion of conditional independence appropri-ate to our domain (conditional separability), we prove that the conditional separabilities induced by a quantum-mechanical wave function are effectively reflected in the graphical struc-ture of MEN. Specifically, we show that for any wave func-tion there exists a MEN which is a perfect map of its condi-tional separabilities. Next, we show how the graphical struc-ture of MEN can be used to effectively classify the pure states of three-qubit systems. We also demonstrate that, in large systems, exploiting conditional independencies may dramat-ically reduce the computational burden of various inference tasks. In principle, the graph-theoretic representation of con-ditional independencies afforded by MEN may not only fa-cilitate the classical simulation of quantum systems, but also provide a guide to the efficient design and complexity analy-sis of quantum algorithms and circuits.
Graphical models of probabilistic dependencies have been extensively investigated in the context of classical uncer-tainty. However, in some domains (most notably, in compu-tational physics and quantum computing) the nature of the relevant uncertainty is non-classical, and the laws of clas-sical probability theory are superseded by those of quan-tum mechanics. In this paper we introduce Markov Entan-glement Networks (MEN), a novel class of graphical repre-sentations of quantum-mechanical dependencies in the con-text of such non-classical systems. MEN are the quantum-mechanical analogue of Markovian Networks, a family of undirected graphical representations which, in the classical domain, exploit a notion of conditional independence among subsystems. After defining a notion of conditional independence appropri-ate to our domain (conditional separability), we prove that the conditional separabilities induced by a quantum-mechanical wave function are effectively reflected in the graphical struc-ture of MEN. Specifically, we show that for any wave func-tion there exists a MEN which is a perfect map of its condi-tional separabilities. Next, we show how the graphical struc-ture of MEN can be used to effectively classify the pure states of three-qubit systems. We also demonstrate that, in large systems, exploiting conditional independencies may dramat-ically reduce the computational burden of various inference tasks. In principle, the graph-theoretic representation of con-ditional independencies afforded by MEN may not only fa-cilitate the classical simulation of quantum systems, but also provide a guide to the efficient design and complexity analy-sis of quantum algorithms and circuits.
Graphical models of probabilistic dependencies have been extensively investigated in the context of classical uncertainty. However, in some domains (most notably, in computational physics and quantum computing) the nature of the relevant uncertainty is non-classical, and the laws of classical probability theory are superseded by those of quantum mechanics. In this paper we introduce Markovian Entanglement Networks (MEN), a novel class of graphical representations of quantum-mechanical dependencies in the context of such non-classical systems. MEN are the quantum-mechanical analogue of Markovian Networks, a family of undirected graphical representations which, in the classical domain, exploit a notion of conditional independence among subsystems. After defining a notion of conditional independence appropriate to our domain (conditional separability), we prove that the conditional separabilities induced by a quantum-mechanical wave function are effectively reflected in the graphical structure of MEN. Specifically, we show that for any wave function there exists a MEN which is a perfect map of its conditional separabilities. Next, we show how the graphical structure of MEN can be used to effectively classify the pure states of three-qubit systems. We also demonstrate that, in large systems, exploiting conditional independencies may dramatically reduce the computational burden of various inference tasks. In principle, the graph-theoretic representation of conditional independencies afforded by MEN may not only facilitate the classical simulation of quantum systems, but also provide a guide to the efficient design and complexity analysis of quantum algorithms and circuits.
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.
We show that any quantum density matrix can be represented by a Bayesian network (a directed acyclic graph), and also by a Markov network (an undirected graph). We show that any Bayesian or Markov net that represents a density matrix, is logically equivalent to a set of conditional independencies (symmetries) satisfied by the density matrix. We show that the d-separation theorems of classical Bayesian and Markov networks generalize in a simple and natural way to quantum physics. The quantum d-separation theorems are shown to be closely connected to quantum entanglement. We show that the graphical rules for d-separation can be used to detect pairs of nodes (or of node sets) in a graph that are unentangled. CMI entanglement (a.k.a. squashed entanglement), a measure of entanglement originally discovered by analyzing Bayesian networks, is an important part of the theory of this paper.
We investigate a correspondence between two formalisms for discrete probabilistic modeling: probabilistic graphical models (PGMs) and tensor networks (TNs), a powerful modeling framework for simulating complex quantum systems. The graphical calculus of PGMs and TNs exhibits many similarities, with discrete undirected graphical models (UGMs) being a special case of TNs. However, more general probabilistic TN models such as Born machines (BMs) employ complex-valued hidden states to produce novel forms of correlation among the probabilities. While representing a new modeling resource for capturing structure in discrete probability distributions, this behavior also renders the direct application of standard PGM tools impossible. We aim to bridge this gap by introducing a hybrid PGM-TN formalism that integrates quantum-like correlations into PGM models in a principled manner, using the physically-motivated concept of decoherence. We first prove that applying decoherence to the entirety of a BM model converts it into a discrete UGM, and conversely, that any subgraph of a discrete UGM can be represented as a decohered BM. This method allows a broad family of probabilistic TN models to be encoded as partially decohered BMs, a fact we leverage to combine the representational strengths of both model families. We experimentally verify the performance of such hybrid models in a sequential modeling task, and identify promising uses of our method within the context of existing applications of graphical models.
Abstract Leveraging the intrinsic probabilistic nature of quantum systems, generative quantum machine learning (QML) offers the potential to outperform classical learning models. Current generative QML algorithms mostly rely on general-purpose models that, while being very expressive, face several training challenges. One potential way to address these setbacks is by constructing problem-informed models that are capable of more efficient training on structured problems. In particular, probabilistic graphical models provide a flexible framework for representing structure in generative learning problems and can thus be exploited to incorporate inductive bias into QML algorithms. In this work, we propose a problem-informed quantum circuit Born machine Ansatz for learning the joint probability distribution of random variables, with independence relations efficiently represented by a Markov network (MN). We further demonstrate the applicability of the MN framework in constructing generative learning benchmarks and compare our model’s performance to previous designs, showing that it outperforms problem-agnostic circuits. Based on a preliminary analysis of trainability, we narrow down the class of MNs to those exhibiting favourable trainability properties. Finally, we discuss the potential of our model to offer quantum advantage in the context of generative learning, and present a practical scenario demonstrating the scalability of the approach.
Tensor-network techniques have enjoyed outstanding success in physics, and have recently attracted attention in machine learning, both as a tool for the formulation of new learning algorithms and for enhancing the mathematical understanding of existing methods. Inspired by these developments, and the natural correspondence between tensor networks and probabilistic graphical models, we provide a rigorous analysis of the expressive power of various tensor-network factorizations of discrete multivariate probability distributions. These factorizations include non-negative tensor-trains/MPS, which are in correspondence with hidden Markov models, and Born machines, which are naturally related to local quantum circuits. When used to model probability distributions, they exhibit tractable likelihoods and admit efficient learning algorithms. Interestingly, we prove that there exist probability distributions for which there are unbounded separations between the resource requirements of some of these tensor-network factorizations. Particularly surprising is the fact that using complex instead of real tensors can lead to an arbitrarily large reduction in the number of parameters of the network. Additionally, we introduce locally purified states (LPS), a new factorization inspired by techniques for the simulation of quantum systems, with provably better expressive power than all other representations considered. The ramifications of this result are explored through numerical experiments. Our findings imply that LPS should be considered over hidden Markov models, and furthermore provide guidelines for the design of local quantum circuits for probabilistic modeling.