arXiv subjects

Davide Pastorello

Publications and source records attributed to Davide Pastorello.

At least 19 recordsLinked to original sources

arXiv2026

Network Security under Heterogeneous Cyber-Risk Profiles and Contagion

Cyber risk has become a critical financial threat in today's interconnected digital economy. This paper introduces a cyber-risk management framework for networked digital systems that combines the strategic behavior of players with contagion dynamics within a security game. We address the problem of optimally allocating cybersecurity resources across a network, focusing on the heterogeneous valuations of nodes by attackers and defenders, some areas may be of high interest to the attacker, while others are prioritized by the defender. We explore how this asymmetry drives attack and defense strategies and shapes the system's overall resilience. We extend a method to determine optimal resource allocation based on simple network metrics weighted by the defender's and attacker's risk profiles. We further propose risk measures based on contagion paths and analyze how propagation dynamics influence optimal defense strategies. Numerical experiments explore risk versus cost efficient frontiers varying network topologies and risk profiles, revealing patterns of resource allocation and cyber deception effects. These findings provide actionable insights for designing resilient digital infrastructures and mitigating systemic cyber risk.

cs.CR↗

arXiv2025

Efficient classical computation of the neural tangent kernel of quantum neural networks

We propose an efficient classical algorithm to estimate the Neural Tangent Kernel (NTK) associated with a broad class of quantum neural networks. These networks consist of arbitrary unitary operators belonging to the Clifford group interleaved with parametric gates given by the time evolution generated by an arbitrary Hamiltonian belonging to the Pauli group. The proposed algorithm leverages a key insight: the average over the distribution of initialization parameters in the NTK definition can be exactly replaced by an average over just four discrete values, chosen such that the corresponding parametric gates are Clifford operations. This reduction enables an efficient classical simulation of the circuit. Combined with recent results establishing the equivalence between wide quantum neural networks and Gaussian processes [Girardi \emph{et al.}, Comm. Math. Phys. 406, 92 (2025); Melchor Hernandez \emph{et al.}, Ann. Henri Poincar{\'e} (2025)], our method enables efficient computation of the expected output of wide, trained quantum neural networks, and therefore shows that such networks cannot achieve quantum advantage.

quant-ph↗

arXiv2025

Enhancing Expressivity of Quantum Neural Networks Based on the SWAP test

Quantum neural networks (QNNs) based on parametrized quantum circuits are promising candidates for machine learning applications, yet many architectures lack clear connections to classical models, potentially limiting their ability to leverage established classical neural network techniques. We examine QNNs built from SWAP test circuits and discuss their equivalence to classical two-layer feedforward networks with quadratic activations under amplitude encoding. Evaluation on real-world and synthetic datasets shows that while this architecture learns many practical binary classification tasks, it has fundamental expressivity limitations: polynomial activation functions do not satisfy the universal approximation theorem, and we show analytically that the architecture cannot learn the parity check function beyond two dimensions, regardless of network size. To address this, we introduce generalized SWAP test circuits with multiple Fredkin gates sharing an ancilla, implementing product layers with polynomial activations of arbitrary even degree. This modification enables successful learning of parity check functions in arbitrary dimensions as well as binary n-spiral tasks, and we provide numerical evidence that the expressivity enhancement extends to alternative encoding schemes such as angle (Z) and ZZ feature maps. We validate the practical feasibility of our proposed architecture by implementing a classically pretrained instance on the IBM Torino quantum processor, achieving 84% classification accuracy on the three-dimensional parity check despite hardware noise. Our work establishes a framework for analyzing and enhancing QNN expressivity through correspondence with classical architectures, and demonstrates that SWAP test-based QNNs possess broad representational capacity relevant to both classical and potentially quantum learning tasks.

quant-ph↗

arXiv2025

A weighted quantum ensemble of homogeneous quantum classifiers

Ensemble methods in machine learning aim to improve prediction accuracy by combining multiple models. This is achieved by ensuring diversity among predictors to capture different data aspects. Homogeneous ensembles use identical models, achieving diversity through different data subsets, and weighted-average ensembles assign higher influence to more accurate models through a weight learning procedure. We propose a method to achieve a weighted homogeneous quantum ensemble using quantum classifiers with indexing registers for data encoding. This approach leverages instance-based quantum classifiers, enabling feature and training point subsampling through superposition and controlled unitaries, and allowing for a quantum-parallel execution of diverse internal classifiers with different data compositions in superposition. The method integrates a learning process involving circuit execution and classical weight optimization, for a trained ensemble execution with weights encoded in the circuit at test-time. Empirical evaluation demonstrate the effectiveness of the proposed method, offering insights into its performance.

quant-ph↗

arXiv2025

Mean-field limit from general mixtures of experts to quantum neural networks

In this work, we study the asymptotic behavior of Mixture of Experts (MoE) trained via gradient flow on supervised learning problems. Our main result establishes the propagation of chaos for a MoE as the number of experts diverges. We demonstrate that the corresponding empirical measure of their parameters is close to a probability measure that solves a nonlinear continuity equation, and we provide an explicit convergence rate that depends solely on the number of experts. We apply our results to a MoE generated by a quantum neural network.

math-ph↗

arXiv2025

Consensus ranking by quantum annealing

Consensus ranking is a technique used to derive a single ranking that best represents the preferences of multiple individuals or systems. It aims to aggregate different rankings into one that minimizes overall disagreement or distance from each of the individual rankings. Kemeny ranking aggregation, in particular, is a widely used method in decision-making and social choice, with applications ranging from search engines to music recommendation systems. It seeks to determine a consensus ranking of a set of candidates based on the preferences of a group of individuals. However, existing quantum annealing algorithms face challenges in efficiently processing large datasets with many candidates. In this paper, we propose a method to improve the performance of quantum annealing for Kemeny rank aggregation. Our approach identifies the pairwise preference matrix that represents the solution list and subsequently reconstructs the ranking using classical methods. This method already yields better results than existing approaches. Furthermore, we present a range of enhancements that significantly improve the proposed method's performance, thereby increasing the number of candidates that can be effectively handled. Finally, we evaluate the efficiency of our approach by comparing its performance and execution time with that of KwikSort, a well-known approximate algorithm.

cs.ET↗

arXiv2024

Quantitative convergence of trained quantum neural networks to a Gaussian process

We study quantum neural networks where the generated function is the expectation value of the sum of single-qubit observables across all qubits. In [Girardi \emph{et al.}, arXiv:2402.08726], it is proven that the probability distributions of such generated functions converge in distribution to a Gaussian process in the limit of infinite width for both untrained networks with randomly initialized parameters and trained networks. In this paper, we provide a quantitative proof of this convergence in terms of the Wasserstein distance of order $1$. First, we establish an upper bound on the distance between the probability distribution of the function generated by any untrained network with finite width and the Gaussian process with the same covariance. This proof utilizes Stein's method to estimate the Wasserstein distance of order $1$. Next, we analyze the training dynamics of the network via gradient flow, proving an upper bound on the distance between the probability distribution of the function generated by the trained network and the corresponding Gaussian process. This proof is based on a quantitative upper bound on the maximum variation of a parameter during training. This bound implies that for sufficiently large widths, training occurs in the lazy regime, \emph{i.e.}, each parameter changes only by a small amount. While the convergence result of [Girardi \emph{et al.}, arXiv:2402.08726] holds at a fixed training time, our upper bounds are uniform in time and hold even as $t \to \infty$.

quant-ph↗

arXiv2024

Quantum concentration inequalities and equivalence of the thermodynamical ensembles: an optimal mass transport approach

We prove new concentration inequalities for quantum spin systems which apply to any local observable measured on any product state or on any state with exponentially decaying correlations. Our results do not require the spins to be arranged in a regular lattice, and cover the case of observables that contain terms acting on spins at arbitrary distance. Moreover, we introduce a local W1 distance, which quantifies the distinguishability of two states with respect to local observables. We prove a transportation-cost inequality stating that the local W1 distance between a generic state and a state with exponentially decaying correlations is upper bounded by a function of their relative entropy. Finally, we apply such inequality to prove the equivalence between the canonical and microcanonical ensembles of quantum statistical mechanics and the weak eigenstate thermalization hypothesis for the Hamiltonians whose Gibbs states have exponentially decaying correlations.

math-ph↗

arXiv2024

Local Binary and Multiclass SVMs Trained on a Quantum Annealer

Support vector machines (SVMs) are widely used machine learning models (e.g., in remote sensing), with formulations for both classification and regression tasks. In the last years, with the advent of working quantum annealers, hybrid SVM models characterised by quantum training and classical execution have been introduced. These models have demonstrated comparable performance to their classical counterparts. However, they are limited in the training set size due to the restricted connectivity of the current quantum annealers. Hence, to take advantage of large datasets (like those related to Earth observation), a strategy is required. In the classical domain, local SVMs, namely, SVMs trained on the data samples selected by a k-nearest neighbors model, have already proven successful. Here, the local application of quantum-trained SVM models is proposed and empirically assessed. In particular, this approach allows overcoming the constraints on the training set size of the quantum-trained models while enhancing their performance. In practice, the FaLK-SVM method, designed for efficient local SVMs, has been combined with quantum-trained SVM models for binary and multiclass classification. In addition, for comparison, FaLK-SVM has been interfaced for the first time with a classical single-step multiclass SVM model (CS SVM). Concerning the empirical evaluation, D-Wave's quantum annealers and real-world datasets taken from the remote sensing domain have been employed. The results have shown the effectiveness and scalability of the proposed approach, but also its practical applicability in a real-world large-scale scenario.

cs.ET↗

arXiv2023

Ensembles of Quantum Classifiers

In the current era, known as Noisy Intermediate-Scale Quantum (NISQ), encoding large amounts of data in the quantum devices is challenging and the impact of noise significantly affects the quality of the obtained results. A viable approach for the execution of quantum classification algorithms is the introduction of a well-known machine learning paradigm, namely, the ensemble methods. Indeed, the ensembles combine multiple internal classifiers, which are characterized by compact sizes due to the smaller data subsets used for training, to achieve more accurate and robust prediction performance. In this way, it is possible to reduce the qubits requirements with respect to a single larger classifier while achieving comparable or improved performance. In this work, we present an implementation and an extensive empirical evaluation of ensembles of quantum classifiers for binary classification, with the purpose of providing insights into their effectiveness, limitations, and potential for enhancing the performance of basic quantum models. In particular, three classical ensemble methods and three quantum classifiers have been taken into account here. Hence, the scheme that has been implemented (in Python) has a hybrid nature. The results (obtained on real-world datasets) have shown an accuracy advantage for the ensemble techniques with respect to the single quantum classifiers, and also an improvement in robustness. In fact, the ensembles have turned out to be able to mitigate both unsuitable data normalizations and repeated measurement inaccuracies, making quantum classifiers more stable.

cs.ET↗

arXiv2023

A general learning scheme for classical and quantum Ising machines

An Ising machine is any hardware specifically designed for finding the ground state of the Ising model. Relevant examples are coherent Ising machines and quantum annealers. In this paper, we propose a new machine learning model that is based on the Ising structure and can be efficiently trained using gradient descent. We provide a mathematical characterization of the training process, which is based upon optimizing a loss function whose partial derivatives are not explicitly calculated but estimated by the Ising machine itself. Moreover, we present some experimental results on the training and execution of the proposed learning model. These results point out new possibilities offered by Ising machines for different learning tasks. In particular, in the quantum realm, the quantum resources are used for both the execution and the training of the model, providing a promising perspective in quantum machine learning.

cs.LG↗

arXiv2023

Classical shadows meet quantum optimal mass transport

Classical shadows constitute a protocol to estimate the expectation values of a collection of M observables acting on O(1) qubits of an unknown n-qubit state with a number of measurements that is independent of n and that grows only logarithmically with M. We propose a local variant of the quantum Wasserstein distance of order 1 of [De Palma et al., IEEE Trans. Inf. Theory 67, 6627 (2021)] and prove that the classical shadow obtained measuring O(log n) copies of the state to be learned constitutes an accurate estimate with respect to the proposed distance. We apply the results to quantum generative adversarial networks, showing that quantum access to the state to be learned can be useful only when some prior information on such state is available.

quant-ph↗

arXiv2023

Scalable quantum neural networks by few quantum resources

This paper focuses on the construction of a general parametric model that can be implemented executing multiple swap tests over few qubits and applying a suitable measurement protocol. The model turns out to be equivalent to a two-layer feedforward neural network which can be realized combining small quantum modules. The advantages and the perspectives of the proposed quantum method are discussed.

quant-ph↗

arXiv2023

A quantum k-nearest neighbors algorithm based on the Euclidean distance estimation

The k-nearest neighbors (k-NN) is a basic machine learning (ML) algorithm, and several quantum versions of it, employing different distance metrics, have been presented in the last few years. Although the Euclidean distance is one of the most widely used distance metrics in ML, it has not received much consideration in the development of these quantum variants. In this article, a novel quantum k-NN algorithm based on the Euclidean distance is introduced. Specifically, the algorithm is characterised by a quantum encoding requiring a low number of qubits and a simple quantum circuit not involving oracles, aspects that favor its realization. In addition to the mathematical formulation and some complexity observations, a detailed empirical evaluation with simulations is presented. In particular, the results have shown the correctness of the formulation, a drop in the performance of the algorithm when the number of measurements is limited, the competitiveness with respect to some classical baseline methods in the ideal case, and the possibility of improving the performance by increasing the number of measurements.

cs.ET↗

arXiv2022

Quantum Annealing Learning Search Implementations

This paper presents the details and testing of two implementations (in C++ and Python) of the hybrid quantum-classical algorithm Quantum Annealing Learning Search (QALS) on a D-Wave quantum annealer. QALS was proposed in 2019 as a novel technique to solve general QUBO problems that cannot be directly represented into the hardware architecture of a D-Wave machine. Repeated calls to the quantum machine within a classical iterative structure and a related convergence proof originate a learning mechanism to find an encoding of a given problem into the quantum architecture. The present work considers the Number Partitioning Problem (NPP) and the Travelling Salesman Problem (TSP) for the testing of QALS. The results turn out to be quite unexpected, with QALS not being able to perform as well as the other considered methods, especially in NPP, where classical methods outperform quantum annealing in general. Nevertheless, looking at the TSP tests, QALS has fulfilled its primary goal, i.e., processing QUBO problems not directly mappable to the QPU topology.

cs.ET↗

arXiv2022

Evaluating the Convergence of Tabu Enhanced Hybrid Quantum Optimization

In this paper we introduce the Tabu Enhanced Hybrid Quantum Optimization metaheuristic approach useful for optimization problem solving on a quantum hardware. We address the theoretical convergence of the proposed scheme from the viewpoint of the collisions in the object which stores the tabu states, based on the Ising model. The results of numerical evaluation of the algorithm on quantum hardware as well as on a classical semiconductor hardware model are also demonstrated.

quant-ph↗

arXiv2022

Implementation and Empirical Evaluation of a Quantum Machine Learning Pipeline for Local Classification

In the current era, quantum resources are extremely limited, and this makes difficult the usage of quantum machine learning (QML) models. Concerning the supervised tasks, a viable approach is the introduction of a quantum locality technique, which allows the models to focus only on the neighborhood of the considered element. A well-known locality technique is the k-nearest neighbors (k-NN) algorithm, of which several quantum variants have been proposed. Nevertheless, they have not been employed yet as a preliminary step of other QML models, whereas the classical counterpart has already proven successful. In this paper, we present (i) an implementation in Python of a QML pipeline for local classification, and (ii) its extensive empirical evaluation. Specifically, the quantum pipeline, developed using Qiskit, consists of a quantum k-NN and a quantum binary classifier. The results have shown the quantum pipeline's equivalence (in terms of accuracy) to its classical counterpart in the ideal case, the validity of locality's application to the QML realm, but also the strong sensitivity of the chosen quantum k-NN to probability fluctuations and the better performance of classical baseline methods like the random forest.

cs.ET↗

arXiv2022

Reconstructing Bayesian Networks on a Quantum Annealer

Bayesian networks are widely used probabilistic graphical models, whose structure is hard to learn starting from the generated data. O'Gorman et al. have proposed an algorithm to encode this task, i.e., the Bayesian network structure learning (BSNL), into a form that can be solved through quantum annealing, but they have not provided an experimental evaluation of it. In this paper, we present (i) an implementation in Python of O'Gorman's algorithm, (ii) a divide et impera approach that allows addressing BNSL problems of larger sizes in order to overcome the limitations imposed by the current architectures, and (iii) their empirical evaluation. Specifically, several problems with an increasing number of variables have been used in the experiments. The results have shown the effectiveness of O'Gorman's formulation for BNSL instances of small sizes, and the superiority of the divide et impera approach on the direct execution of O'Gorman's algorithm.

cs.ET↗