School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371 Inferring causal models from observed correlations is a challenging task, crucial to many areas of science. In order to alleviate the effort, it is important to know whether symmetries in the observations correspond to symmetries in the underlying realization. Via an explicit example, we answer this question in the negative. We use a tripartite probability distribution over binary events that is realized by using three (different) independent sources of classical randomness. We prove that even removing the condition that the sources distribute systems described by classical physics, the requirements that i) the sources distribute the same physical systems, ii) these physical systems respect relativistic causality, and iii) the correlations are the observed ones, are incompatible.

We present a tensorization algorithm for constructing tensor train representations of functions, drawing on sketching and cross interpolation ideas. The method only requires black-box access to the target function and a small set of sample points defining the domain of interest. Thus, it is particularly well-suited for machine learning models, where the domain of interest is naturally defined by the training dataset. We show that this approach can be used to enhance the privacy and interpretability of neural network models. Specifically, we apply our decomposition to (i) obfuscate neural networks whose parameters encode patterns tied to the training data distribution, and (ii) estimate topological phases of matter that are easily accessible from the tensor train representation. Additionally, we show that this tensorization can serve as an efficient initialization method for optimizing tensor trains in general settings, and that, for model compression, our algorithm achieves a superior trade-off between memory and time complexity compared to conventional tensorization methods of neural networks.

It is widely known that Boltzmann machines are capable of representing arbitrary probability distributions over the values of their visible neurons, given enough hidden ones. However, sampling -- and thus training -- these models can be numerically hard. Recently we proposed a regularisation of the connections of Boltzmann machines, in order to control the energy landscape of the model, paving a way for efficient sampling and training. Here we formally prove that such regularised Boltzmann machines preserve the ability to represent arbitrary distributions. This is in conjunction with controlling the number of energy local minima, thus enabling easy \emph{guided} sampling and training. Furthermore, we explicitly show that regularised Boltzmann machines can store exponentially many arbitrarily correlated visible patterns with perfect retrieval, and we connect them to the Dense Associative Memory networks.

Tensor networks, widely used for providing efficient representations of low-energy states of local quantum many-body systems, have been recently proposed as machine learning architectures which could present advantages with respect to traditional ones. In this work we show that tensor network architectures have especially prospective properties for privacy-preserving machine learning, which is important in tasks such as the processing of medical records. First, we describe a new privacy vulnerability that is present in feedforward neural networks, illustrating it in synthetic and real-world datasets. Then, we develop well-defined conditions to guarantee robustness to such vulnerability, which involve the characterization of models equivalent under gauge symmetry. We rigorously prove that such conditions are satisfied by tensor-network architectures. In doing so, we define a novel canonical form for matrix product states, which has a high degree of regularity and fixes the residual gauge that is left in the canonical forms based on singular value decompositions. We supplement the analytical findings with practical examples where matrix product states are trained on datasets of medical records, which show large reductions on the probability of an attacker extracting information about the training dataset from the model's parameters. Given the growing expertise in training tensor-network architectures, these results imply that one may not have to be forced to make a choice between accuracy in prediction and ensuring the privacy of the information processed.

Tensor networks are factorizations of high-dimensional tensors into network-like structures composed of smaller tensors. Originating from condensed matter physics and acclaimed for their efficient representation of quantum many-body systems [1-10], these structures have allowed researchers to comprehend the intricate properties of such systems and, additionally, simulate them using classical computers [11-13]. Notably, tensor networks are the most successful method for simulating the results of quantum advantage experiments [14-16]. Furthermore, tensor networks were rediscovered within the numerical linear algebra community [17-19], where the techniques have been adapted to other high-dimensional problems such as numerical integration [20], signal processing [21], or epidemic modelling [22]. With the advent of machine learning and the the quest for expressive yet easy-to-train models, tensor networks have been suggested as promising candidates, due to their ability to parameterize regions of the complex space of size exponential in the number of input features. Since the pioneering works [23, 24] that used simple, 1-dimensional networks known as Matrix Product States (MPS) in the physics literature [4, 25] and as Tensor Trains in the numerical linear algebra literature [18], these have been applied in both supervised and unsupervised learning settings [26-28].

Binary Neural Networks are a promising technique for implementing efficient deep models with reduced storage and computational requirements. The training of these is however, still a compute-intensive problem that grows drastically with the layer size and data input. At the core of this calculation is the linear regression problem. The Harrow-Hassidim-Lloyd (HHL) quantum algorithm has gained relevance thanks to its promise of providing a quantum state containing the solution of a linear system of equations. The solution is encoded in superposition at the output of a quantum circuit. Although this seems to provide the answer to the linear regression problem for the training neural networks, it also comes with multiple, difficult-to-avoid hurdles. This paper shows, however, that useful information can be extracted from the quantum-mechanical implementation of HHL, and used to reduce the complexity of finding the solution on the classical side.

Bayesian methods in machine learning, such as Gaussian processes, have great advantages compared to other techniques. In particular, they provide estimates of the uncertainty associated with a prediction. Extending the Bayesian approach to deep architectures has remained a major challenge. Recent results connected deep feedforward neural networks with Gaussian processes, allowing training without backpropagation. This connection enables us to leverage a quantum algorithm designed for Gaussian processes and develop new algorithms for Bayesian deep learning on quantum computers. The properties of the kernel matrix in the Gaussian process ensure the efficient execution of the core component of the protocol, quantum matrix inversion, providing a polynomial speedup with respect to classical algorithm. Furthermore, we demonstrate the execution of the algorithm on contemporary quantum computers and analyze its robustness to realistic noise models.