Archivio 2025 382 seminari

Title: Log/local quasi-map correspondence Abstract: I will explain an equivalence between the virtual count of rational curves in the total space of an anti-nef line bundle and the virtual count of rational curves maximally tangent to a smooth section of the dual line bundle due to Graber--van Garrel--Ruddat. I will present generalisations which involve quasi-maps. This is work with A. Cobos-Rabano and Q Shafi.

Derived enumerative geometry Abstract: Derived algebraic geometry provides a powerful set of tools to enumerative geometers, giving geometric spaces which encode the "virtual structures" of the moduli problems . I will discuss a joint work with D. Karn, E. Mann and C. Manolache in which we define a derived enhancement for the moduli space of sections.

Title: Terminal 3-folds that are not Cohen-Macaulay Abstract: An important local vanishing theorem for the minimal model program is the fact that klt singularities in characteristic zero are Cohen-Macaulay. In contrast, even in the narrow setting of terminal singularities of dimension 3, we show that Cohen-Macaulayness can fail in characteristic p or mixed characteristic (0,p) for p equal to 2, 3, or 5. This is optimal, by work of Arvidsson-Bernasconi-Lacini. (These examples help to explain why the MMP remains an open question for 3-folds in characteristic 2 or 3.) The examples are quotients of regular schemes by the cyclic group G of order p. In characteristic p or mixed characteristic (0,p), such quotients can exhibit a wide range of behavior.

Title: Tableaux Littlewood—Richardson rules for 2-step flags Abstract: The Abelian/non-Abelian correspondence gives rise to a natural basis for the cohomology of flag varieties, which - except for Grassmannians - is distinct from the Schubert basis. I will describe this basis and its multiplication rules, and explain how to relate it to the Schubert basis for two-step flag varieties. I will then explain how this leads to new tableaux Littlewood--Richardson rules for many products of Schubert classes. This is joint work (separately) with Wei Gu and Linda Chen.

Title: The Noether-Lefschetz locus in families of threefolds Abstract: For a fixed threefold X and very ample line bundle H, the Noether-Lefschetz locus parametrizes surfaces in X which are linearly equivalent to H and have Picard rank greater than that of X. We discuss the behavior of special components of the Noether-Lefschetz locus as we deform the pair (X, H), particularly when X is a singular Fano threefold. This is joint work with A. Grassi and J. Rana.

The study of Jacobians and their compactifications is central to understanding the geometry of algebraic curves, especially in the context of degenerations and moduli. Compactified Jacobians provide essential tools for extending classical theorems and constructions to singular curves, revealing deep connections between algebraic geometry, combinatorics, and tropical geometry. This mini-course will explore these rich interactions. We will begin with the theory of compactified Jacobians for nodal curves. We will then introduce tropical Jacobians, showing how they arise combinatorially from metric graphs. A central focus will be the Abel map for nodal curves and its resolution, highlighting the interplay between algebraic and tropical geometry. Finally, we will discuss the Torelli theorem for nodal curves, addressing the question of reconstructing a curve from its compactified Jacobian. The course aims to provide a bridge between the classical geometric theory of Jacobians and the modern perspectives offered by tropical geometry and degeneration techniques.

Tropical geometry studies a piecewise linear combinatorial shadow of degenerations and compactifications of algebraic varieties. A typical phenomenon is that many of the usual algebro-geometric objects have a tropical analogue that is intimately tied to its classical counterpart. An example is the theory of divisors and line bundles on algebraic curves, whose tropical counterparts have been crucial in numerous surprising applications to classical Brill—Noether theory and the birational geometry of moduli spaces. One classical object that has resisted the effort of tropical geometers so far is the geometry of vector bundles beyond rank one. In this talk, I will outline an elementary approach to tropical vector bundles that builds on earlier work of Allermann. Although limited in scope, this theory leads to a satisfying tropical story for semistable vector bundles on elliptic curves and, more generally, semihomogeneous vector bundles on abelian varieties. The engines in the background that make these cases accessible to our methods are Atiyah's classification of vector bundles on elliptic curves, Fourier-Mukai transforms on abelian varieties, and the interactions with non-Archimedean uniformization. This talk is based on joint work with Andreas Gross and Dmitry Zakharov (and, in parts, Arne Kuhrs) as well as with Andreas Gross, Inder Kaur, and Annette Werner.

Severi varieties parametrize integral curves of fixed geometric genus in a given linear system on a surface. In this talk, I will present some results on the irreducibility of these varieties in the case of toric surfaces, and their application to the irreducibility of other moduli spaces of curves. This is done using tropical methods and I will indicate some of these aspects. The new results are from ongoing joint work with Xiang He and Ilya Tyomkin.

I will explain several new ideas leading to a better understanding of the cohomology of moduli spaces of (stable) curves of all genera. Joint with Hannah Larson, Sam Payne, and Thomas Willwacher.

Diffusion models have achieved remarkable success across a wide range of generative tasks. A key challenge is understanding the mechanisms that prevent their memorization of training data and allow generalization. In this seminar I will discuss the role of the training dynamics in the transition from generalization to memorization. I will show the emergence of two distinct timescales: an early time $\tau_\mathrm{gen}$ at which models begin to generate high-quality samples, and a later time $\tau_\mathrm{mem}$ beyond which memorization emerges. Crucially, we find that $\tau_\mathrm{mem}$ increases linearly with the training set size $n$, while $\tau_\mathrm{gen}$ remains constant. This creates a growing window of training times with $n$ where models generalize effectively, despite showing strong memorization if training continues beyond it. It is only when $n$ becomes larger than a model-dependent threshold that overfitting disappears at infinite training times. These findings reveal a form of implicit dynamical regularization in the training dynamics, which allow to avoid memorization even in highly overparameterized settings. Our results are supported by numerical experiments with standard U-Net architectures on realistic and synthetic datasets, and by a theoretical analysis using a tractable random features model studied in the high-dimensional limit.

Deep generative models parametrize very flexible families of distributions able to fit complicated datasets of images or text. These models provide independent samples from complex high-distributions at negligible costs. On the other hand, sampling exactly a target distribution, such as the Boltzmann distribution of a physical system, is typically challenging: either because of dimensionality, multi-modality, ill-conditioning or a combination of the previous. In this talk, I will discuss opportunities and challenges in enhancing traditional Monte Carlo methods with learning.

We theoretically investigate the phenomena of generalization and memorization in diffusion models. Empirical studies suggest that these phenomena are influenced by model complexity and the size of the training dataset. In our experiments, we further observe that the number of noise samples per data sample (m) used during Denoising Score Matching (DSM) plays a significant and non-trivial role. We capture these behaviors and shed insights into their mechanisms by deriving asymptotically precise expressions for test and train errors of DSM under a simple theoretical setting. The score function is parameterized by random features neural networks, with the target distribution being d-dimensional Gaussian. We operate in a regime where the dimension d, number of data samples n, and number of features p tend to infinity while keeping the ratios n/d and p/d fixed. By characterizing the test and train errors, we identify regimes of generalization and memorization as a function of n/d, p/d, and m. Our theoretical findings are consistent with the empirical observations.

Abstraction is the process of extracting the essential features from raw data while ignoring irrelevant details. This is similar to the process of focusing on large-scale properties, systematically removing irrelevant small-scale details, implemented in the renormalisation group of statistical physics. This analogy is suggestive because the fixed points of the renormalisation group offer an ideal candidate of a truly abstract -- i.e. data independent -- representation. It has been observed that abstraction emerges with depth in neural networks. Deep layers of neural network capture abstract characteristics of data, such as "cat-ness" or "dog-ness" in images, by combining the lower level features encoded in shallow layers (e.g. edges). Yet we argue that depth alone is not enough to develop truly abstract representations. We advocate that the level of abstraction crucially depends on how broad the training set is. We address the issue within a renormalisation group approach where a representation is expanded to encompass a broader set of data. We take the unique fixed point of this transformation -- the Hierarchical Feature Model -- as a candidate for an abstract representation. This theoretical picture is tested in numerical experiments based on Deep Belief Networks trained on data of different breadth. These show that representations in deep layers of neural networks approach the Hierarchical Feature Model as the data gets broader, in agreement with theoretical predictions.

Stochastic differential equations (SDE) driven by white noise are important models for stochastic dynamical systems in natural science and engineering. The statistical inference of the parameters of such models based on noisy observations has also attracted considerable interest in the machine learning community. Using Girsanov's change of measure approach one can apply powerful variational techniques to solve the inference problem. A limitation of standard SDE models is the fact that they show typically a fast, exponential decay of correlation functions. If one is interested in stochastic processes with a long time memory, a well known possibility is to replace the Brownian motion in the SDE by the so called fractional Brownian motion (fBM) which is no longer a Markov process. Unfortunately, variational inference for this case is much less straightforward. Our approach to this problem utilises a somewhat overlooked idea by Carmona and Coutin (1998) who showed that fBM can be exactly represented as an infinite dimensional linear combination of Ornstein-Uhlenbeck processes with different time constants. Using an appropriate discretisation, we arrive at a finite dimensional approximation which is an 'ordinary' SDE model in an augmented space. For this new model we can apply (more or less) off-the shelve variational inference approaches. We also discuss application of this approach to generative diffusion models.

Understanding the limit of algorithms in solving hard optimization combinatorial problems is a fundamental problem both in basic research and real-world applications. However, results are scarce, especially for sparse models, which are the most realistic ones. I will summarize our current understanding of algorithmic thresholds in well-known problems, like satisfiability and coloring, focusing mainly on the dependence of algorithmic thresholds on the time scaling and on the analytical attempts to estimate these thresholds.

Low-degree polynomials provide a versatile methodology to build systematic approximations of high-dimensional inference problems. In this talk I will present some recent results obtained by applying this framework to problems of supervised learning, focussing in particular on two layers neural networks of extensive width.

We investigate the impact of limited data on training pairwise energy-based models for inverse problems aimed at identifying interaction networks. Utilizing the Gaussian model as testbed, we dissect training trajectories across the eigenbasis of the coupling matrix, exploiting the independent evolution of eigenmodes and revealing that the learning timescales are tied to the spectral decomposition of the empirical covariance matrix. We see that optimal points for early stopping arise from the interplay between these timescales and the initial conditions of training. Moreover, we show that finite data corrections can be accurately modeled through asymptotic random matrix theory calculations and provide the counterpart of generalized cross-validation in the energy based model context. Our analytical framework extends to binary-variable maximum-entropy pairwise models with minimal variations. These findings offer strategies to control overfitting in discrete-variable models through empirical shrinkage corrections, improving the management of overfitting in energy-based generative models. Finally, we propose a generalization to arbitrary energy-based models by deriving the neural tangent kernel dynamics of the score function under the score-matching algorithm.

Understanding the generalization properties of large, overparametrized, neural networks is a central problem in theoretical machine learning. Several insightful ideas have been proposed in this regard, among them: the implicit regularization hypothesis, the possibility of having benign overfitting and the existence of feature learning regimes where neural networks learn the latent structure of data. However a precise understanding of the emergence/validity of these behaviors cannot be disentangled from the study of the non-linear training dynamics. We use a technique from statistical physics, dynamical mean field theory, to study the training dynamics and obtain a rich picture of how generalization and overfitting arise in large overparametrized models. In particular, focusing on large 2-layer neural networks, we point out: (i) the emergence of a separation of timescales controlling feature learning and overfitting, (ii) a non-monotone behavior of the test error and, correspondingly, a 'feature unlearning' phase at large times and (iii) the emergence of algorithmic inductive bias towards small complexity. Joint work with Andrea Montanari.

Boltzmann distribution in physics, Gibbs measure in mathematics, exponential family distributions in statistics, undirected graphical models in computer science, or energy-based models in machine learning — all these notions refer to the same general concept, also known as Markov Random Fields (MRFs). A recurrent interest for MRFs in many different branches of science is explained by the fact that they serve as a natural and interpretable modeling foundation for many scientific applications: MRFs have been used for modeling of natural systems at equilibrium since the creation of statistical physics! Yet, unknown training algorithms, as well as the lack of tools for generating predictions from these models presented the main barriers to the widespread use of MRFs in Scientific Machine Learning. In this talk, we review the state-of-the-art for learning of MRFs from data, and for constructing MRFs in forms which allow for an efficient generation of predictions and sampling. We illustrate a wide applicability of this concept in several distinct scientific areas: random graph models, statistical and quantum mechanical models, and field theories.

Recent advances in natural language processing highlight two key factors for improving reasoning in large language models (LLMs): (i) allocating more test-time compute tends to help on harder problems but often introduces redundancy in the reasoning trace, and (ii) compute is most effective when reasoning is systematic and incremental, forming structured chains of thought (CoTs) akin to human problem-solving. To study these factors in isolation, we introduce a controlled setting based on shortest-path tasks in layered graphs. We train decoder-only transformers on question–trace–answer triples using a custom tokenizer, comparing models trained on optimal bottom-up dynamic programming traces with those trained on longer, valid traces involving backtracking. Surprisingly, with the same training-token budget, models trained on inefficient traces generalize better to unseen graphs. This benefit is not due to length alone—injecting arbitrary redundancy into reasoning traces fails to help and can even hurt performance. Instead, we find that generalization correlates with the model's confidence in next-token prediction, suggesting that long, coherent, and locally incremental traces make the training signal easier to optimize.

We study the computational hardness of finding collisions in neural networks, i.e. any two sets of weights that give the same labels to a random dataset. When the number of output neurons is sufficiently large, we establish the emergence of an overlap gap in the space of collisions. This is believed to indicate that efficient algorithms will not be able to find collisions. Such claim is supported by numerical experiments using approximate message passing algorithms, which stop working below the predicted value from the analysis. Our results also show that, by composing such a collision resistant neural network with an Error Correcting Code, one can obtain a Hash Function. Beyond relevance to cryptography for designing collision resistant one-way functions, our work uncovers new forms of computational hardness emerging in large neural networks.

Recent years have seen substantial advances in our understanding of high-dimensional ridge regression, but existing theories assume that training examples are independent. By leveraging techniques from random matrix theory and free probability, we provide sharp asymptotics for the in- and out-of-sample risks of ridge regression when the data points have arbitrary correlations. We demonstrate that in this setting, the generalized cross validation estimator (GCV) fails to correctly predict the out-of-sample risk. However, in the case where the noise residuals have the same correlations as the data points, one can modify the GCV to yield an efficiently-computable unbiased estimator that concentrates in the high-dimensional limit, which we dub CorrGCV. We further extend our asymptotic analysis to the case where the test point has nontrivial correlations with the training set, a setting often encountered in time series forecasting. Assuming knowledge of the correlation structure of the time series, this again yields an extension of the GCV estimator, and sharply characterizes the degree to which such test points yield an overly optimistic prediction of long-time risk.

The task of sampling efficiently the Gibbs-Boltzmann distribution of disordered systems is important both for the theoretical understanding of these models and for the solution of practical optimization problems. Unfortunately, this task is known to be hard, especially for spin glasses at low temperatures. Recently, many attempts have been made to tackle the problem by mixing classical Monte Carlo schemes with newly devised Neural Networks that learn to propose smart moves. In this talk I will review a few physically-interpretable deep architectures, and in particular one whose number of parameters scales linearly with the size of the system and that can be applied to a large variety of topologies. I will show that these architectures can accurately learn the Gibbs-Boltzmann distribution for the two-dimensional and three-dimensional Edwards-Anderson models, and specifically for some of its most difficult instances. I will show that the performance increases with the number of layers, in a way that clearly connects to the correlation length of the system, thus providing a simple and interpretable criterion to choose the optimal depth. Finally, I will discuss the performances of these architectures in proposing smart Monte Carlo moves and compare to state-of-the-art algorithms.

Learning tasks are playing an increasingly central role in quantum information and computation from fundamental problems like state discrimination and metrology to quantum PAC learning and the recently proposed “shadow” variants of state tomography. Yet these various strands of quantum learning theory have largely evolved in isolation. In this talk, we introduce a unified mathematical framework for quantum learning based on classical–quantum training data and show how to bound a quantum learner’s expected generalization error on new data. Our bounds are expressed in terms of classical and quantum information theoretic quantities that capture how strongly the learner’s hypothesis depends on the specific training data. To derive them, we develop non commutative analogues of the decoupling lemmas underlying recent classical information theoretic generalization bounds, drawing on tools from quantum optimal transport and quantum concentration inequalities. This framework subsumes and yields intuitive generalization bounds for a variety of quantum learning scenarios including quantum state discrimination, PAC learning of quantum states or classical functions, and quantum parameter estimation laying the groundwork for a unified, information theoretic perspective on quantum learning.

We consider the problem of testing and learning from data in the presence of resource constraints, such as limited memory or weak data access, which place limitations on the efficiency and feasibility of testing or learning. In particular, we ask the following question: Could a resource-constrained learner/tester use interaction with a resource-unconstrained but untrusted party to solve a learning or testing problem more efficiently than they could without such an interaction? In this work, we answer this question both abstractly and for concrete problems, in two complementary ways: For a wide variety of scenarios, we prove that a resource-constrained learner cannot gain any advantage through classical interaction with an untrusted prover. As a special case, we show that for the vast majority of testing and learning problems in which quantum memory is a meaningful resource, a memory-constrained quantum algorithm cannot overcome its limitations via classical communication with a memory unconstrained quantum prover. In contrast, when quantum communication is allowed, we construct a variety of interactive proof protocols, for specific learning and testing problems, which allow memory constrained quantum verifiers to gain significant advantages through delegation to untrusted provers. These results highlight both the limitations and potential of delegating learning and testing problems to resource-rich but untrusted third parties.

Uhlmann's theorem states that, for any two quantum states ρAB and σA, there exists an extension σAB of σA such that the fidelity between ρAB and σAB equals the fidelity between their reduced states ρA and σA. In this work, we generalize Uhlmann's theorem to α-Rényi relative entropies for α∈[12,∞], a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to α=12, α=1, and α=∞, respectively. Based on joint work with Giulia Mazzola and Renato Renner.

Simulating quantum circuits classically is in general a hard task. However, certain families of quantum circuits may be practically or even provably efficiently simulable by use of specialized classical algorithms. In this talk, we will cover "Pauli propagation" which has recently been shown to enable efficient classical simulation of expectation values in quantum circuits and a wide range of noise-free quantum circuits. Appreciating the strengths and weaknesses of this simulation method, and how it can be efficiently combined with other classical and quantum subroutines, will help point towards promising applications of quantum devices. We will end by discussing a generalization of this approach to Fermionic systems opening up new applications in quantum chemistry and material science. This talk will give an overview of the following works: arxiv:2308.09109, arXiv:2408.12739, arXiv:2409.01706, arXiv:2411.19896, arXiv:2501.13101, arXiv:2503.18939.

Entanglement is a fundamental resource in quantum computation and quantum communication, but it is potentially affected by decoherence phenomena that make it necessary to introduce appropriate tests to certify and quantify the degree of entanglement of a quantum state. In this talk. I will show how a photonic integrated circuit designed to implement the swap test algorithm can be adapted to an efficient entanglement witness for both pure and mixed bipartite states.

Motivated by realistic hardware considerations of the pre-fault-tolerant era, we comprehensively study the impact of uncorrected noise on typical quantum circuits. We first show that any noise `truncates’ most quantum circuits to effectively logarithmic depth, in the task of estimating observable expectation values. We then prove that quantum circuits under any non-unital noise exhibit lack of barren plateaus for cost functions composed of local observables. But, we also design an efficient classical algorithm to estimate observable expectation values of typical quantum circuits within any target constant accuracy, in any circuit architecture. Taken together, our results showcase that, unless we carefully engineer the circuits to take advantage of the noise, it is unlikely that noisy quantum circuits provide any quantum advantage for algorithms that output observable expectation value estimates, like many variational quantum machine learning proposals.

Classical neural networks with random initialization famously behave as Gaussian processes in the limit of many neurons, which allows one to completely characterize their training and generalization behavior. While there are settings where quantum neural networks (QNNs) have also been shown to behave as Gaussian processes, there exist known counterexamples to this behavior. We here prove that QNNs and their first two derivatives instead generally form what we call "Wishart processes," where certain algebraic properties of the network determine the hyperparameters of the process. This Wishart process description allows us to, for the first time: give necessary and sufficient conditions for a QNN architecture to have a Gaussian process limit; calculate the full gradient distribution, generalizing previously known barren plateau results; and calculate the local minima distribution of algebraically constrained QNNs. Our unified framework suggests a certain simple operational definition for the "trainability" of a given QNN model using a newly introduced, experimentally accessible quantity we call the "degrees of freedom" of the network architecture.

We explore the broad question posed in the title from different perspectives. We show how a classical neural network can be trained to optimally embed features into a quantum system and to optimally extract information from it. We review the concept of variational quantum error mitigation, i.e., the idea of variationally optimizing error mitigation strategies. We present recent results demonstrating how classical deep learning models and noisy quantum computers can cooperate to better estimate quantum expectation values. Finally, as a speculative open problem, we propose pushing the core question to its extreme limit: Can AI autonomously decide how to optimally use a noisy quantum computer without hard-coding any specific error-reduction strategy?

In this talk, I will present two recent works related to the question of quantum advantages in machine learning. In the first work, we address a major obstacle to the widespread use of quantum machine learning models in practice: quantum 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 class of quantum models where quantum resources are only required during training, while the deployment of the trained model is classical. 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. In the second work, we expand our understanding of where quantum advantages can be found in quantum machine learning, by showing a PAC learning advantage in the realm of shallow-depth circuits. This learning advantage has the particularity that it is unconditional, meaning that we do not need to make assumptions such as the existence of classically hard, quantumly easy, cryptographic functions to show an advantage. The machine learning task we consider is that of learning probability distributions, or generative learning. We design this learning task building on recent results by Bene Watts and Parham on quantum advantages for sampling, which we technically uplift to a hyperplane learning problem, identifying non-local correlations as the origin of the quantum advantage.

Quantum Machine Learning (QML) has recently been explored as a novel approach to surpass the capabilities of classical methods, although the field remains in its early stages and the outcomes achieved so far are still inconclusive. This talk offers a critical and methodologically grounded perspective on current QML approaches, with particular attention to the fundamental limitations of classical machine learning and the ways in which quantum-enhanced models may be designed to address these challenges. Recent developments in unsupervised and reinforcement learning serve as illustrative examples to examine how quantum formulations, tailored to the structure of specific problems, can yield algorithmic and representational advantages. Methodological aspects such as model design, problem encoding, and hybrid integration are emphasized, along with a discussion of current limitations in quantum computing, including hardware constraints and the lack of mature, task-specific quantum design principles. The talk concludes with reflections on how these insights may inform the development of more robust and effective QML methodologies.

Investigating the input-output relations of a quantum process can be seen as a learning problem. For instance, we could wish to find the optimal parameters for some quantum device that let it best mimic our target process, or we could simply wish to construct the best possible mathematical model of the process. Experimentally, we send probes through the quantum channel and use the outputs as our training set. When learning the action of a continuous variable (CV) quantum process in this way, there will often be some restriction on the input states used. One experimentally simple way to probe CV channels is using low-energy coherent states. Learning a quantum channel in this way presents difficulties, since two channels may act similarly on low energy inputs but very differently for high energy inputs. They may also act similarly on coherent state inputs but differently on non-classical inputs. Extrapolating the behaviour of a channel for more general input states from its action on the far more limited set of low energy coherent states is a case of out-of-distribution generalisation. To be sure that such generalisation gives meaningful results, one needs to relate error bounds for the training set to bounds that are valid for all inputs. We show that for any pair of channels that act sufficiently similarly on low energy coherent state inputs, one can bound how different the input-output relations are for any (high energy or highly non-classical) input. This proves out-of-distribution generalisation is always possible for learning quantum channels using low energy coherent states.

Simulating arbitrary quantum dynamics with classical algorithms is widely believed to be intractable. Yet, by exploiting the structure of certain restricted settings, specialized classical methods can succeed. One particularly promising family - Pauli propagation - recasts simulation in the Pauli basis and often delivers rigorous runtime and error guarantees. At the heart of these guarantees lie two ingredients: the presence of local noise, which dampens long-range interactions, and a degree of “scrambling” in the circuit’s gates. In this talk, we will present our recent results on applying Pauli propagation to both noisy and noiseless circuits. Along the way, we’ll discuss to what extent noise and scrambling influence simulability - and what that tells us about the necessary resources for quantum advantage.

We study quantum neural networks made by parametric one-qubit gates and fixed two-qubit gates in the limit of infinite width, where the generated function is the expectation value of the sum of single-qubit observables over all the qubits. First, we prove that the probability distribution of the function generated by the untrained network with randomly initialized parameters converges to a Gaussian process whenever each measured qubit is correlated only with few other measured qubits. Then, we analytically characterize the training of the network via gradient descent with square loss on supervised learning problems. In particular, as long as the network is not affected by barren plateaus, the trained network can perfectly fit the training set and that the probability distribution of the function generated after training still converges in distribution to a Gaussian process, also in the presence of the statistical noise of the measurement at the output of the network. For finite size circuits, we make the convergence quantitative in terms of the Wasserstein distance of order 1.

Italy is at the forefront of shaping the European HPC-QC ecosystem, playing a key role in two major initiatives: EuroQHPC-I and QEC4QEA. As one of the selected hosting entities for a European quantum computer, Italy is set to pioneer the integration of quantum computing with high-performance computing (HPC). This integration, conducted alongside other selected hosting entities, will mark a significant step toward the hybrid computing architectures of the future. Simultaneously, Italy has been chosen to lead Europe’s first Center of Excellence in Quantum Computing, QEC4QEA. This initiative will drive the development of the first HPC-QC applications, accelerating the adoption of quantum technologies in scientific and industrial domains. By spearheading both infrastructure deployment and software innovation, Italy is in pole position to build the future European HPC-QC ecosystem, reinforcing its leadership in quantum and high-performance computing.

Cat or dog? Can a Hong-Ou-Mandel interferometer tell the difference? This talk presents a quantum optical method for binary classification that recognizes patterns without the need for image reconstruction. By encoding both data and model parameters into single-photon states and leveraging two-photon interference, the system classifies patterns directly through coincidence rates. Acting as a quantum analogue of a classical neuron, it operates—once trained—with constant O(1) resource complexity, achieving a superexponential speedup over its classical counterpart.

This talk is about a quantitative stability result for geodesic spheres in rank-one symmetric spaces of non-compact type — including real, complex, quaternionic, and octonionic hyperbolic spaces. These spaces have negatively pinched sectional curvature, whose minima is distributed according to the underlying algebraic structure. This geometric framework, and in particular the distribution on the tangent space associated with any radial vector fields, plays a central role in the analysis. We show that geodesic spheres are uniformly stable under small volume-preserving $C^1$-perturbations, with perimeter gain controlled by the $W^{1,2}$-norm of the perturbation. As a consequence, we give a quantitative proof that, for small volumes, geodesic spheres are the unique isoperimetric regions.

We present the first examples of nonsmooth sub-Riemannian length minimizing curves. The length minimizer with the lowest regularity within these examples is of class $C^2\setminus C^3$. The singularity is at a boundary point. The result is sharp in the sense that we can prove that, within these examples, it is not possible to find a minimizer of class $C^1\setminus C^2$. This is a joint work with Y. Chitour, F. Jean, R. Monti, L. Rifford, L. Sacchelli, and M. Sigalotti.