Density operators, quantum generalizations of probability distributions, are gaining prominence in machine learning due to their foundational role in quantum computing. Generative modeling based on density operator models (\textbf{DOMs}) is an emerging field, but existing training algorithms -- such as those for the Quantum Boltzmann Machine -- do not scale to real-world data, such as the MNIST dataset. The Expectation-Maximization algorithm has played a fundamental role in enabling scalable training of probabilistic latent variable models on real-world datasets. \textit{In this paper, we develop an Expectation-Maximization framework to learn latent variable models defined through \textbf{DOMs} on classical hardware, with resources comparable to those used for probabilistic models, while scaling to real-world data.} However, designing such an algorithm is nontrivial due to the absence of a well-defined quantum analogue to conditional probability, which complicates the Expectation step. To overcome this, we reformulate the Expectation step as a quantum information projection (QIP) problem and show that the Petz Recovery Map provides a solution under sufficient conditions. Using this formulation, we introduce the Density Operator Expectation Maximization (DO-EM) algorithm -- an iterative Minorant-Maximization procedure that optimizes a quantum evidence lower bound. We show that the \textbf{DO-EM} algorithm ensures non-decreasing log-likelihood across iterations for a broad class of models. Finally, we present Quantum Interleaved Deep Boltzmann Machines (\textbf{QiDBMs}), a \textbf{DOM} that can be trained with the same resources as a DBM. When trained with \textbf{DO-EM} under Contrastive Divergence, a \textbf{QiDBM} outperforms larger classical DBMs in image generation on the MNIST dataset, achieving a 40--60\% reduction in the Fréchet Inception Distance.

Distances between probability distributions are a key component of many statistical machine learning tasks, from two-sample testing to generative modeling, among others. We introduce a novel distance between measures that compares them through a Schatten norm of their kernel covariance operators. We show that this new distance is an integral probability metric that can be framed between a Maximum Mean Discrepancy (MMD) and a Wasserstein distance. In particular, we show that it avoids some pitfalls of MMD, by being more discriminative and robust to the choice of hyperparameters. Moreover, it benefits from some compelling properties of kernel methods, that can avoid the curse of dimensionality for their sample complexity. We provide an algorithm to compute the distance in practice by introducing an extension of kernel matrix for difference of distributions that could be of independent interest. Those advantages are illustrated by robust approximate Bayesian computation under contamination as well as particle flow simulations.

Despite the many challenges in exploratory data analysis, artificial neural networks have motivated strong interests in scientists and researchers both in theoretical as well as practical applications. Among sources of such popularity of artificial neural networks the ability of modeling non-linear dynamical systems, generalization, and adaptation possibilities should be mentioned. Despite this, there is still significant debate about the role of various underlying stochastic processes in stabilizing a unique structure for data learning and prediction. One of such obstacles to the theoretical and numerical study of machine intelligent systems is the curse of dimensionality and the sampling from high-dimensional probability distributions. In general, this curse prevents efficient description of states, providing a significant complexity barrier for the system to be efficiently described and studied. In this strand of research, direct treatment and description of such abstract notions of learning theory in terms of quantum information be one of the most favorable candidates. Hence, the subject matter of these articles is devoted to problems of design, adaptation and the formulations of computationally hard problems in terms of quantum mechanical systems. In order to characterize the microscopic description of such dynamics in the language of inferential statistics, covariance matrix estimation of d-dimensional Gaussian densities and Bayesian interpretation of eigenvalue problem for dynamical systems is assessed.

We provide a decision-theoretic framework for dealing with uncertainty in quantum mechanics. This uncertainty is two-fold: on the one hand there may be uncertainty about the state the quantum system is in, and on the other hand, as is essential to quantum mechanical uncertainty, even if the quantum state is known, measurements may still produce an uncertain outcome. In our framework, measurements therefore play the role of acts with an uncertain outcome and our simple decision-theoretic postulates ensure that Born's rule is encapsulated in the utility functions associated with such acts. This approach allows us to uncouple (precise) probability theory from quantum mechanics, in the sense that it leaves room for a more general, so-called imprecise probabilities approach. We discuss the mathematical implications of our findings, which allow us to give a decision-theoretic foundation to recent seminal work by Benavoli, Facchini and Zaffalon, and we compare our approach to earlier and different approaches by Deutsch and Wallace.

We can learn (more) about the state a quantum system is in through measurements. We look at how to describe the uncertainty about a quantum system's state conditional on executing such measurements. We show that by exploiting the interplay between desirability, coherence and indifference, a general rule for conditioning can be derived. We then apply this rule to conditioning on measurement outcomes, and show how it generalises to conditioning on a set of measurement outcomes.

Given a Boolean formula and a functions assigning weights to assignments of values to the Boolean variable, we consider the problems of Weighted Constrained Sampling (WCS) and Weighted Model Counting (WMC). The first, also called distributionaware sampling (Chakraborty et al, 2014), involves sampling assignments to the Boolean variables with a probability proportional to their weight given that the formula is satisfied. The latter (Sang et al, 2005) consists in computing the sum of the weights of the models of the formula, i.e. the weighted model count. WCS has important applications in a variety of domanis, including statistical physics (Jerrum and Sinclair, 1996), statistics (Madras and Piccioni, 1999), hardware verification (Naveh et al, 2006), and probabilistic reasoning, where it can be used to solve the problem of Most Probable Explanation (MPE) and Maximum A Posteriori (MAP). MPE (Sang et al, 2007) involves finding an assignment to all variables that satisfies a Boolean formula and has the maximum weight. The related MAP problem means finding an assignment of a subset of the variables such that the sum of the weights of the models of the formula that agree on the assignment is maximum. WMC was successfully applied, among others, to the problem of performing inference in graphical models (Chavira and Darwiche, 2008; Sang et al, 2005).

I present a quick and sound method for the robustness verification of a sort of quantum classifiers who are Linear Sound. Since quantum machine learning has been put into practice in relevant fields and Linear Sound Property, LSP is a pervasive property, the method could be universally applied. I implemented my method with a Quantum Convolutional Neural Network, QCNN using MindQuantum, Huawei and successfully verified its robustness when classifying MNIST dataset.

Density operators representing ensembles of pure states of sample wave functions are used in place probability densities. We show that such representation allows to formulate the statistical Bayesian learning problem in different coordinate systems on the sample space. We further show that such representation allows to learn projections of density operators using a kernel trick. In particular, the study highlights that decomposing wave functions rather than probability densities, as it is done in kernel embedding, allows to preserve the nature of probability operators. Results are illustrated with a simple example using discrete orthogonal wavelet transform of density operators.

The emergence of variational quantum applications has led to the development of automatic differentiation techniques in quantum computing. Recently, Zhu et al. (PLDI 2020) have formulated differentiable quantum programming with bounded loops, providing a framework for scalable gradient calculation by quantum means for training quantum variational applications. However, promising parameterized quantum applications, e.g., quantum walk and unitary implementation, cannot be trained in the existing framework due to the natural involvement of unbounded loops. To fill in the gap, we provide the first differentiable quantum programming framework with unbounded loops, including a newly designed differentiation rule, code transformation, and their correctness proof. Technically, we introduce a randomized estimator for derivatives to deal with the infinite sum in the differentiation of unbounded loops, whose applicability in classical and probabilistic programming is also discussed. We implement our framework with Python and Q#, and demonstrate a reasonable sample efficiency. Through extensive case studies, we showcase an exciting application of our framework in automatically identifying close-to-optimal parameters for several parameterized quantum applications.

Two types of states are widely used in quantum mechanics, namely (deterministic-coefficient) pure states and statistical mixtures. A density operator can be associated with each of them. We here address a third type of states, that we previously introduced in a more restricted framework. These states generalize pure ones by replacing each of their deterministic ket coefficients by a random variable. We therefore call them Random-Coefficient Pure States, or RCPS. We analyze their properties and their relationships with both types of usual states. We show that RCPS contain much richer information than the density operator and mean of observables that we associate with them. This occurs because the latter operator only exploits the second-order statistics of the random state coefficients, whereas their higher-order statistics contain additional information. That information can be accessed in practice with the multiple-preparation procedure that we propose for RCPS, by using second-order and higher-order statistics of associated random probabilities of measurement outcomes. Exploiting these higher-order statistics opens the way to a very general approach for performing advanced quantum information processing tasks. We illustrate the relevance of this approach with a generic example, dealing with the estimation of parameters of a quantum process and thus related to quantum process tomography. This parameter estimation is performed in the non-blind (i.e. supervised) or blind (i.e. unsupervised) mode. We show that this problem cannot be solved by using only the density operator \rho of an RCPS and the associated mean value Tr(\rho A) of the operator A that corresponds to the considered physical quantity. We succeed in solving this problem by exploiting a fourth-order statistical parameter of state coefficients, in addition to second-order statistics. Numerical tests validate this result.