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Predicting quantum operator matrices such as Hamiltonian, overlap, and density matrices in the density functional theory (DFT) framework is crucial for understanding material properties. Current methods often focus on individual operators and struggle with efficiency and scalability for large systems. Here we introduce a novel deep learning model, SLEM (Strictly Localized Equivariant Message-passing) for predicting multiple quantum operators, that achieves state-of-the-art accuracy while dramatically improving computational efficiency. SLEM's key innovation is its strict locality-based design, constructing local, equivariant representations for quantum tensors while preserving physical symmetries. This enables complex many-body dependence without expanding the effective receptive field, leading to superior data efficiency and transferability. Using an innovative SO(2) convolution technique, SLEM reduces the computational complexity of high-order tensor products and is therefore capable of handling systems requiring the $f$ and $g$ orbitals in their basis sets. We demonstrate SLEM's capabilities across diverse 2D and 3D materials, achieving high accuracy even with limited training data. SLEM's design facilitates efficient parallelization, potentially extending DFT simulations to systems with device-level sizes, opening new possibilities for large-scale quantum simulations and high-throughput materials discovery.

We present a new algorithm for solving linear-quadratic regulator (LQR) problems with linear equality constraints. This is the first such exact algorithm that is guaranteed to have a runtime that is linear in the number of stages, as well as linear in the number of both state-only constraints as well as mixed state-and-control constraints, without imposing any restrictions on the problem instances. We also show how to easily parallelize this algorithm to run in parallel runtime logarithmic in the number of stages of the problem.

Quantum machine learning is often highlighted as one of the most promising practical applications for which quantum computers could provide a computational advantage. However, a major obstacle to the widespread use of quantum machine learning models in practice is that these models, even once trained, still require access to a quantum computer in order to be evaluated on new data. To solve this issue, we introduce a new class of quantum models where quantum resources are only required during training, while the deployment of the trained model is classical. Specifically, the training phase of our models ends with the generation of a 'shadow model' from which the classical deployment becomes possible. We prove that: i) this class of models is universal for classically-deployed quantum machine learning; ii) it does have restricted learning capacities compared to 'fully quantum' models, but nonetheless iii) it achieves a provable learning advantage over fully classical learners, contingent on widely-believed assumptions in complexity theory. These results provide compelling evidence that quantum machine learning can confer learning advantages across a substantially broader range of scenarios, where quantum computers are exclusively employed during the training phase. By enabling classical deployment, our approach facilitates the implementation of quantum machine learning models in various practical contexts.

For factor analysis, many estimators, starting with the maximum likelihood estimator, have been developed, and the statistical properties of most estimators have been well explored. In the early 2000s, a new estimator based on matrix factorization, called Matrix Decomposition Factor Analysis (MDFA), was developed. Although the estimator is obtained by minimizing the principal component analysis-like loss function, this estimator empirically behaves like other consistent estimators of factor analysis, not principal component analysis. Since the MDFA estimator cannot be formulated as a classical M-estimator, the statistical properties of the MDFA estimator have yet to be discussed. To explain this unexpected behavior theoretically, we establish the consistency of the MDFA estimator for factor analysis. That is, we show that the MDFA estimator converges to the same limit as other consistent estimators of factor analysis.

Quantum machine learning models based on parametrized quantum circuits, also called quantum neural networks (QNNs), are considered to be among the most promising candidates for applications on near-term quantum devices. Here we explore the expressivity and inductive bias of QNNs by exploiting an exact mapping from QNNs with inputs $x$ to classical perceptrons acting on $x \otimes x$ (generalised to complex inputs). The simplicity of the perceptron architecture allows us to provide clear examples of the shortcomings of current QNN models, and the many barriers they face to becoming useful general-purpose learning algorithms. For example, a QNN with amplitude encoding cannot express the Boolean parity function for $n\geq 3$, which is but one of an exponential number of data structures that such a QNN is unable to express. Mapping a QNN to a classical perceptron simplifies training, allowing us to systematically study the inductive biases of other, more expressive embeddings on Boolean data. Several popular embeddings primarily produce an inductive bias towards functions with low class balance, reducing their generalisation performance compared to deep neural network architectures which exhibit much richer inductive biases. We explore two alternate strategies that move beyond standard QNNs. In the first, we use a QNN to help generate a classical DNN-inspired kernel. In the second we draw an analogy to the hierarchical structure of deep neural networks and construct a layered non-linear QNN that is provably fully expressive on Boolean data, while also exhibiting a richer inductive bias than simple QNNs. Finally, we discuss characteristics of the QNN literature that may obscure how hard it is to achieve quantum advantage over deep learning algorithms on classical data.

In this paper, we propose a new numerical strategy for the stabilization of evolution systems. The method is based on the methodology given by Ammari, Nicaise and Pignotti in "Stabilization by switching time-delay, Asymptot. Anal., 83 (2013), 263--283". This method is then implemented in 1D by suitable numerical approximation techniques. Numerical experiments complete this study to confirm the theoretical announced results.

Meta-learning, or "learning to learn," is a subfield of machine learning where the goal is to develop models and algorithms that can learn from various tasks and improve their learning process over time. Unlike traditional machine learning methods focusing on learning a specific task, meta-learning aims to leverage experience from previous tasks to enhance future learning. This approach is particularly beneficial in scenarios where the available data for a new task is limited, but there exists abundant data from related tasks. By extracting and utilizing the underlying structure and patterns across these tasks, meta-learning algorithms can achieve faster convergence and better performance with fewer data. The following notes are mainly inspired from \cite{vanschoren2018meta}, \cite{baxter2019learning}, and \cite{maurer2005algorithmic}.

We present a theory of "quantum references", similar to lenses in classical functional programming, that allow to point to a subsystem of a larger quantum system, and to mutate/measure that part. Mutable classical variables, quantum registers, and wires in quantum circuits are examples of this, but also references to parts of larger quantum datastructures. Quantum references in our setting can also refer to subparts of other references, or combinations of parts from different references, or quantum references seen in a different basis, etc. Our modeling is intended to be well suited for formalization in theorem provers and as a foundation for modeling variables in quantum programs. We study quantum references in greater detail and cover the infinite-dimensional case as well, but also provide a more general treatment not specific to the quantum case. We implemented a large part of our results (including a small quantum Hoare logic and an analysis of quantum teleportation) in the Isabelle/HOL theorem prover.

This work explores multi-modal inference in a high-dimensional simplified model, analytically quantifying the performance gain of multi-modal inference over that of analyzing modalities in isolation. We present the Bayes-optimal performance and weak recovery thresholds in a model where the objective is to recover the latent structures from two noisy data matrices with correlated spikes. The paper derives the approximate message passing (AMP) algorithm for this model and characterizes its performance in the high-dimensional limit via the associated state evolution. The analysis holds for a broad range of priors and noise channels, which can differ across modalities. The linearization of AMP is compared numerically to the widely used partial least squares (PLS) and canonical correlation analysis (CCA) methods, which are both observed to suffer from a sub-optimal recovery threshold.

A new, more efficient, numerical method for the SDOF problem is presented. Its construction is based on the weak form of the equation of motion, as obtained in part I of the paper, using piece-wise polynomial functions as interpolation functions. The approximation rate can be arbitrarily high, proportional to the degree of the interpolation functions, tempered only by numerical instability. Moreover, the mechanical energy of the system is conserved. Consequently, all significant drawbacks of existing algorithms, such as the limitations imposed by the Dahlqvist Barrier theorem and the need for introduction of numerical damping, have been overcome.