Comparing probability distributions is a core challenge across the natural, social, and computational sciences. Existing methods, such as Maximum Mean Discrepancy (MMD), struggle in high-dimensional and non-compact domains. Here we introduce quantum probability metrics (QPMs), derived by embedding probability measures in the space of quantum states: positive, unit-trace operators on a Hilbert space. This construction extends kernel-based methods and overcomes the incompleteness of MMD on non-compact spaces. Viewed as an integral probability metric (IPM), QPMs have dual functions that uniformly approximate all bounded, uniformly continuous functions on $\mathbb{R}^n$, offering enhanced sensitivity to subtle distributional differences in high dimensions. For empirical distributions, QPMs are readily calculated using eigenvalue methods, with analytic gradients suited for learning and optimization. Although computationally more intensive for large sample sizes ($O(n^3)$ vs. $O(n^2)$), QPMs can significantly improve performance as a drop-in replacement for MMD, as demonstrated in a classic generative modeling task. By combining the rich mathematical framework of quantum mechanics with classical probability theory, this approach lays the foundation for powerful tools to analyze and manipulate probability measures.

Continuous-variable quantum key distribution [1] is designed to implement point-to-point secret key distribution, its security is guaranteed by the fundamental laws of quantum physics [2]. In a basic version of CVQKD [3], the sender, called Alice, encodes secret key bits in the phase space of coherent states and sends them to an insecure quantum channel, while the receiver, called Bob, measures these incoming signal states with coherent detection. After several steps of data post-processing, a string of secret keys can be finally shared by Alice and Bob. One of the advantages of CVQKD is that it is compatible with most existing commercial telecommunication technologies [4-6], making it easier to integrate into real-world communication links. In general, a CVQKD system is mainly composed of quantum signal processing and data-postprocessing [7]. The former part corresponds to signal modulation, transmission, and measurement, aiming to generate a raw key, while the latter part corresponds to data reconciliation, parameter estimation, and privacy amplification, attempting to extract the final secret key from the raw key. In CVQKD, secret key rate and maximal transmission distance are generally a pair of crucial performance indicators. For a specific CVQKD system, however, there is tradeoff between the secret key rate and the maximal transmission distance: the longer the transmission distance, the lower the secret key rate, and vice versa. The main reason is that the continuous-variable quantum signal used to carry the secret key is extremely weak.

Variational quantum circuits (VQCs) [1] are candidates for achieving practical quantum advantages in the noisy intermediate-scale quantum (NISQ) era [2], when scalable error-corrected quantum computers are not yet available. VQCs utilize classical control to optimize a quantum circuit to solve computation problems, including optimization [3], eigen-system problem [4-10], partial-differential equations [11], quantum simulation [12-14] and machine learning [15-23]. As a general approach of designing quantum circuits, it has also found applications in the approximation [24], preparation [25, 26], classification [27-31] and tomography [32] of quantum states. Initial works on VQCs concern discrete-variable (DV) finite-dimensional systems of qubits, which are natural for computation; while continous-variable (CV) systems of bosonic qumodes are less explored. Yet, many important quantum systems are modelled by qumodes. For example, quantum communication and networking [33-37] rely on photons--the only flying quantum information carrier. In this regard, quantum transduction and entanglement distillation are shown to be enhanced by CV VQCs [38]; Photonic quantum computers [39, 40] are also relying on bosonic encoding such as the cat code and Gottesman-Kitaev-Preskill (GKP) code [41], which has shown great promise [42, 43]. The engineering of such code states are greatly boosted by CV VQCs [44-47]; Finally, distributed entangled sensor networks ubiquitously rely on CV VQCs to achieve quantum advantages in sensing [48-51] and data classification [52, 53]. Different from traditional algorithms, the runtime of VQCs is characterized by the time necessary to train the variational parameters to optimize a cost function.

Experimental sciences have come to depend heavily on our ability to organize and interpret high-dimensional datasets. Natural laws, conservation principles, and inter-dependencies among observed variables yield geometric structure, with fewer degrees of freedom, on the dataset. We introduce the frameworks of semiclassical and microlocal analysis to data analysis and develop a novel, yet natural uncertainty principle for extracting fine-scale features of this geometric structure in data, crucially dependent on data-driven approximations to quantum mechanical processes underlying geometric optics. This leads to the first tractable algorithm for approximation of wave dynamics and geodesics on data manifolds with rigorous probabilistic convergence rates under the manifold hypothesis. We demonstrate our algorithm on real-world datasets, including an analysis of population mobility information during the COVID-19 pandemic to achieve four-fold improvement in dimensionality reduction over existing state-of-the-art and reveal anomalous behavior exhibited by less than 1.2% of the entire dataset. Our work initiates the study of data-driven quantum dynamics for analyzing datasets, and we outline several future directions for research.

We miss general methods for quantum solitons, although they can act as entanglement generators or as self-organized quantum processors. We develop a computational approach that uses a neural network as a variational ansatz for quantum solitons in an array of waveguides. By training the resulting phase space quantum machine learning model, we find different soliton solutions varying the number of particles and interaction strength. We consider Gaussian states that enable measuring the degree of entanglement and sampling the probability distribution of many-particle events. We also determine the probability of generating particle pairs and unveil that soliton bound states emit correlated pairs. These results may have a role in boson sampling with nonlinear systems and in quantum processors for entangled nonlinear waves. A soliton is a non-perturbative solution of a classical nonlinear wave-equation; it may describe mean-field states of atoms (as in Bose-Einstein condensation) or photons (as in nonlinear optics) [1]. From a quantum mechanical perspective, a soliton may correspond to a coherent state; however, the nonlinearity may induce squeezing or non-Gaussianity [2]. The quantum properties of solitons inspired experimental investigations, as quantum non-demolition, squeezing [3-6] and photon bound states [7]. Authors reported on theoretical studies on the soliton quantum features, as evaporation and breathing [8-13].

We introduce an algorithm for computing geodesics on sampled manifolds that relies on simulation of quantum dynamics on a graph embedding of the sampled data. Our approach exploits classic results in semiclassical analysis and the quantum-classical correspondence, and forms a basis for techniques to learn the manifold from which a dataset is sampled, and subsequently for nonlinear dimensionality reduction of high-dimensional datasets. We illustrate the new algorithm with data sampled from model manifolds and also by a clustering demonstration based on COVID-19 mobility data. Finally, our method reveals interesting connections between the discretization provided by data sampling and quantization.

Quantum hypothesis testing is one of the most fundamental problems in quantum information theory, with crucial implications in areas like quantum sensing, where it has been used to prove quantum advantage in a series of binary photonic protocols, e.g., for target detection or memory cell readout. In this work, we generalize this theoretical model to the multi-partite setting of barcode decoding and pattern recognition. We start by defining a digital image as an array or grid of pixels, each pixel corresponding to an ensemble of quantum channels. Specializing each pixel to a black and white alphabet, we naturally define an optical model of barcode. In this scenario, we show that the use of quantum entangled sources, combined with suitable measurements and data processing, greatly outperforms classical coherent-state strategies for the tasks of barcode data decoding and classification of black and white patterns. Moreover, introducing relevant bounds, we show that the problem of pattern recognition is significantly simpler than barcode decoding, as long as the minimum Hamming distance between images from different classes is large enough. Finally, we theoretically demonstrate the advantage of using quantum sensors for pattern recognition with the nearest neighbor classifier, a supervised learning algorithm, and numerically verify this prediction for handwritten digit classification.

We introduce a genetic algorithm that designs quantum optics experiments for engineering quantum states with specific properties. Our algorithm is powerful and flexible, and can easily be modified to find methods of engineering states for a range of applications. Here we focus on quantum metrology. First, we consider the noise-free case, and use the algorithm to find quantum states with a large quantum Fisher information (QFI). We find methods, which only involve experimental elements that are available with current technology, for engineering quantum states with up to a 100-fold improvement over the best classical state, and a 20-fold improvement over the optimal Gaussian state. Such states are a superposition of the vacuum with a large number of photons (around 80), and can hence be seen as Schr\"odinger-cat-like states. We then apply the two most dominant noise sources in our setting -- photon loss and imperfect heralding -- and use the algorithm to find quantum states that still improve over the optimal Gaussian state with realistic levels of noise. This will open up experimental and technological work in using exotic non-Gaussian states for quantum-enhanced phase measurements. Finally, we use the Bayesian mean square error to look beyond the regime of validity of the QFI, finding quantum states with precision enhancements over the alternatives even when the experiment operates in the regime of limited data.

IKERBASQUE, Basque F oundation for Science, Mar ıa D ıaz de Haro 3, E-48013 Bilbao, Spain We propose a method to build quantum memristors in quantum photonic platforms. We firstly design an effective beam splitter, which is tunable in real-time, by means of a Mach-Zehnder-type array with two equal 50:50 beam splitters and a tunable retarder, which allows us to control its reflectivity. Then, we show that this tunable beam splitter, when equipped with weak measurements and classical feedback, behaves as a quantum memristor. Indeed, in order to prove its quantumness, we show how to codify quantum information in the coherent beams. Moreover, we estimate the memory capability of the quantum memristor.

The most important requirements, for an operator to be viewed as a proposition, is that it must be hermitian and idempotent (which, in the Hilbert case corresponds to projectors). We interpret the above restrictions as follows. Hermitian operators have real eigenvalues. In particular, idempotent operators have eigenvalues 0 or 1, that is, they allow for asserting or negating in the classical way. When the operator is not hermitian, it is true that there is no way to interpret it directly as a logical proposition, because its eigenvalues are not real numbers, and the proposition cannot be asserted as usual.