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Environmental phenomena are influenced by complex interactions among various factors. For instance, the amount of rainfall measured at different stations within a given area is shaped by atmospheric conditions, orography, and physics of water processes. Motivated by the need to analyze rainfall across complex spatial locations, we propose a flexible Bayesian semi-parametric model for spatially distributed data. This method effectively accounts for spatial correlation while incorporating dependencies on geographical characteristics in a highly flexible manner. Indeed, using latent Gaussian processes, indexed by spatial coordinates and topographical features, the model integrates spatial dependencies and environmental characteristics within a nonparametric framework. Posterior inference is conducted using an efficient rejection-free Markov Chain Monte Carlo algorithm, which eliminates the need for tuning parameter calibration, ensuring smoother and more reliable estimation. The model's flexibility is evaluated through a series of simulation studies, involving different rainfall and spatial correlation scenarios, to demonstrate its robustness across various conditions. We then apply the model to a large dataset of rainfall events collected from the Italian regions of Veneto and Trentino-Alto Adige, these areas are known for their complex orography and diverse meteorological drivers. By analyzing this data, we generate detailed maps that illustrate the mean and variance of rainfall and rainy days. The method is implemented in a new R package available on GitHub.

If $X\subset\operatorname{Gr}(2,6)$ is the Fano variety of lines of a smooth cubic fourfold, then we show that the restriction to $X$ of any Schur functor of the tautological quotient bundle is modular and slope polystable. Moreover it is atomic if and only if it is rigid, in which case it is also slope stable. We further compute the Ext-groups of such bundles in infinitely many cases, showing in particular the existence of new modular vector bundles on manifolds of type $\operatorname{K3}^{[2]}$ that are slope stable and whose $\operatorname{Ext}^1$-group is 40-dimensional., Comment: Comments are very welcome!

Randomized measurements are increasingly appreciated as powerful tools to estimate properties of quantum systems, e.g., in the characterization of hybrid classical-quantum computation. On many platforms they constitute natively accessible measurements, serving as the building block of prominent schemes like shadow estimation. In the real world, however, the implementation of the random gates at the core of these schemes is susceptible to various sources of noise and imperfections, strongly limiting the applicability of protocols. To attenuate the impact of this shortcoming, in this work we introduce an error-mitigated method of randomized measurements, giving rise to a robust shadow estimation procedure. On the practical side, we show that error mitigation and shadow estimation can be carried out using the same session of quantum experiments, hence ensuring that we can address and mitigate the noise affecting the randomization measurements. Mathematically, we develop a picture derived from Fourier-transforms to connect randomized benchmarking and shadow estimation. We prove rigorous performance guarantees and show the functioning using comprehensive numerics. More conceptually, we demonstrate that, if properly used, easily accessible data from randomized benchmarking schemes already provide such valuable diagnostic information to inform about the noise dynamics and to assist in quantum learning procedures., Comment: 6+20 pages, 6 figures

Understanding textual description to generate code seems to be an achieved capability of instruction-following Large Language Models (LLMs) in zero-shot scenario. However, there is a severe possibility that this translation ability may be influenced by having seen target textual descriptions and the related code. This effect is known as Data Contamination. In this study, we investigate the impact of Data Contamination on the performance of GPT-3.5 in the Text-to-SQL code-generating tasks. Hence, we introduce a novel method to detect Data Contamination in GPTs and examine GPT-3.5's Text-to-SQL performances using the known Spider Dataset and our new unfamiliar dataset Termite. Furthermore, we analyze GPT-3.5's efficacy on databases with modified information via an adversarial table disconnection (ATD) approach, complicating Text-to-SQL tasks by removing structural pieces of information from the database. Our results indicate a significant performance drop in GPT-3.5 on the unfamiliar Termite dataset, even with ATD modifications, highlighting the effect of Data Contamination on LLMs in Text-to-SQL translation tasks.

The combination of quantum many-body and machine learning techniques has recently proved to be a fertile ground for new developments in quantum computing. Several works have shown that it is possible to classically efficiently predict the expectation values of local observables on all states within a phase of matter using a machine learning algorithm after learning from data obtained from other states in the same phase. However, existing results are restricted to phases of matter such as ground states of gapped Hamiltonians and Gibbs states that exhibit exponential decay of correlations. In this work, we drop this requirement and show how it is possible to learn local expectation values for all states in a phase, where we adopt the Lindbladian phase definition by Coser \& P\'erez-Garc\'ia [Coser \& P\'erez-Garc\'ia, Quantum 3, 174 (2019)], which defines states to be in the same phase if we can drive one to other rapidly with a local Lindbladian. This definition encompasses the better-known Hamiltonian definition of phase of matter for gapped ground state phases, and further applies to any family of states connected by short unitary circuits, as well as non-equilibrium phases of matter, and those stable under external dissipative interactions. Under this definition, we show that $N = O(\log(n/\delta)2^{polylog(1/\epsilon)})$ samples suffice to learn local expectation values within a phase for a system with $n$ qubits, to error $\epsilon$ with failure probability $\delta$. This sample complexity is comparable to previous results on learning gapped and thermal phases, and it encompasses previous results of this nature in a unified way. Furthermore, we also show that we can learn families of states which go beyond the Lindbladian definition of phase, and we derive bounds on the sample complexity which are dependent on the mixing time between states under a Lindbladian evolution., Comment: 19 pages, 3 figures, 21 page appendix

In this paper we study monodromy operators on moduli spaces $M_v(S,H)$ of sheaves on K3 surfaces with non-primitive Mukai vectors $v$. If we write $v=mw$, with $m>1$ and $w$ primitive, then our main result is that the inclusion $M_w(S,H)\to M_v(S,H)$ as the most singular locus induces an isomorphism between the monodromy groups of these symplectic varieties, allowing us to extend to the non-primitive case a result of Markman., Comment: 52 pages, comments are very welcome!

Cheap-to-Build Very Large-Language Models (CtB-LLMs) with affordable training are emerging as the next big revolution in natural language processing and understanding. These CtB-LLMs are democratizing access to trainable Very Large-Language Models (VLLMs) and, thus, may represent the building blocks of many NLP systems solving downstream tasks. Hence, a little or a large bias in CtB-LLMs may cause huge harm. In this paper, we performed a large investigation of the bias of three families of CtB-LLMs, and we showed that debiasing techniques are effective and usable. Indeed, according to current tests, the LLaMA and the OPT families have an important bias in gender, race, religion, and profession. In contrast to the analysis for other LLMs, we discovered that bias depends not on the number of parameters but on the perplexity. Finally, the debiasing of OPT using LoRA reduces bias up to 4.12 points in the normalized stereotype score.

Understanding how to characterise and mitigate errors is a key challenge in developing reliable quantum architecture for near-term applications. Recent work (arXiv:2103.17243) provides an efficient set of algorithms for analysing unknown noise processes requiring only tomographic snapshots of the quantum operator under consideration, without the need of any a-priori information on the noise model, nor necessitating a particular experimental setup. The only assumption made is that the observed channel can be approximated by a time-independent Markovian map, which is a typically reasonable framework when considering short time scales. In this note we lift the time-independent assumption, presenting an extension of the scheme now able to analyse noisy dynamics with time-dependent generators from a sequence of snapshots. We hence provide a diagnostic tool for a wider spectrum of instances while inheriting all the favourable features from the previous protocol. On the theoretical side, the problem of characterising time-dependent Markovian channels has been an open problem for many decades. This work gives an approach to tackle this characterisation problem rigorously., Comment: 15 pages

We consider the general problem of Bayesian binary regression and we introduce a new class of distributions, the Perturbed Unified Skew Normal (pSUN, henceforth), which generalizes the Unified Skew-Normal (SUN) class. We show that the new class is conjugate to any binary regression model, provided that the link function may be expressed as a scale mixture of Gaussian densities. We discuss in detail the popular logit case, and we show that, when a logistic regression model is combined with a Gaussian prior, posterior summaries such as cumulants and normalizing constants can be easily obtained through the use of an importance sampling approach, opening the way to straightforward variable selection procedures. For more general priors, the proposed methodology is based on a simple Gibbs sampler algorithm. We also claim that, in the p > n case, the proposed methodology shows better performances - both in terms of mixing and accuracy - compared to the existing methods. We illustrate the performance through several simulation studies and two data analyses.

We discuss an acceptance-rejection algorithm for the random number generation from the Kolmogorov distribution. Since the cumulative distribution function (CDF) is expressed as a series, in order to obtain the density function we need to prove that the series of the derivatives converges uniformly. We also provide a similar proof in order to show that the ratio between the target Kolmogorov density and the auxiliary density implemented is bounded. Finally we discuss a way of truncating the series expression of the density in an optimal way.

We prove that any symplectic automorphism of finite order of an irreducible holomorphic symplectic manifold of O'Grady's 10-dimensional deformation type is trivial., Comment: 8 pages, accepted on Proc. Amer. Math. Soc

We study infinitesimal deformations of autodual and hyper-holomorphic connections on complex vector bundles on hyper-K\"ahler manifolds of arbitrary dimension. In particular, we describe the DG Lie algebra controlling this deformation problem. Moreover, we prove associative formality for derived endomorphisms of a holomorphic vector bundle admitting a projectively hyper-holomorphic connection., Comment: 26 pages. Version 2: exposition improved, some changes to Sections 3 and 5, geometric applications to projectively hyper-holomorphic vector bundles unchanged

We extend classical results of Perego and Rapagnetta on moduli spaces of sheaves of type OG10 to moduli spaces of Bridgeland semistable objects on the Kuznetsov component of a cubic fourfold. In particular, we determine the period of this class of varieties and use it to understand when they become birational to moduli spaces of sheaves on a K3 surface., Comment: 21 pages, comments are welcome. v2: Minor changes. To appear in Canadian Journal of Mathematics

We discuss a Bayesian hierarchical copula model for clusters of financial time series. A similar approach has been developed in recent paper. However, the prior distributions proposed there do not always provide a proper posterior. In order to circumvent the problem, we adopt a proper global-local shrinkage prior, which is also able to account for potential dependence structures among different clusters. The performance of the proposed model is presented via simulations and a real data analysis.

Pre-trained Transformers are challenging human performances in many NLP tasks. The massive datasets used for pre-training seem to be the key to their success on existing tasks. In this paper, we explore how a range of pre-trained Natural Language Understanding models perform on definitely unseen sentences provided by classification tasks over a DarkNet corpus. Surprisingly, results show that syntactic and lexical neural networks perform on par with pre-trained Transformers even after fine-tuning. Only after what we call extreme domain adaptation, that is, retraining with the masked language model task on all the novel corpus, pre-trained Transformers reach their standard high results. This suggests that huge pre-training corpora may give Transformers unexpected help since they are exposed to many of the possible sentences.

With quantum computing devices increasing in scale and complexity, there is a growing need for tools that obtain precise diagnostic information about quantum operations. However, current quantum devices are only capable of short unstructured gate sequences followed by native measurements. We accept this limitation and turn it into a new paradigm for characterizing quantum gate-sets. A single experiment - random sequence estimation - solves a wealth of estimation problems, with all complexity moved to classical post-processing. We derive robust channel variants of shadow estimation with close-to-optimal performance guarantees and use these as a primitive for partial, compressive and full process tomography as well as the learning of Pauli noise. We discuss applications to the quantum gate engineering cycle, and propose novel methods for the optimization of quantum gates and diagnosing cross-talk., Comment: 10+17 pages, 3 figures, replaced with final version

The presence of noise is currently one of the main obstacles to achieving large-scale quantum computation. Strategies to characterise and understand noise processes in quantum hardware are a critical part of mitigating it, especially as the overhead of full error correction and fault-tolerance is beyond the reach of current hardware. Non-Markovian effects are a particularly unfavourable type of noise, being both harder to analyse using standard techniques and more difficult to control using error correction. In this work we develop a set of efficient algorithms, based on the rigorous mathematical theory of Markovian master equations, to analyse and evaluate unknown noise processes. In the case of dynamics consistent with Markovian evolution, our algorithm outputs the best-fit Lindbladian, i.e., the generator of a memoryless quantum channel which best approximates the tomographic data to within the given precision. In the case of non-Markovian dynamics, our algorithm returns a quantitative and operationally meaningful measure of non-Markovianity in terms of isotropic noise addition. We provide a Python implementation of all our algorithms, and benchmark these on a range of 1- and 2-qubit examples of synthesised noisy tomography data, generated using the Cirq platform. The numerical results show that our algorithms succeed both in extracting a full description of the best-fit Lindbladian to the measured dynamics, and in computing accurate values of non-Markovianity that match analytical calculations., Comment: published version, 36 + 24 pages, merging material from arXiv:2303.08936. Code available at: https://gitlab.com/TamaraKohler/non-markovianity

The experimental thermal neutron cross sections of the twenty proteinogenic amino acids have been measured over the incident-neutron energy range spanning from 1 meV to 10 keV and data have been interpreted using the multi-phonon expansion based on first-principles calculations. The scattering cross section, dominated by the incoherent inelastic contribution from the hydrogen atoms, can be rationalised in terms of the average contributions of different functional groups, thus neglecting their correlation. These results can be used for modelling the total neutron cross sections of complex organic systems like proteins, muscles, or human tissues from a limited number of starting input functions. This simplification is of crucial importance for fine-tuning of transport simulations used in medical applications, including boron neutron capture therapy as well as secondary neutrons-emission induced during proton therapy. Moreover, the parametrized neutron cross sections allow a better treatment of neutron scattering experiments, providing detailed sample self-attenuation corrections for a variety of biological and soft-matter systems.

Renormalisation group (RG) methods provide one of the most important techniques for analysing the physics of many-body systems, both analytically and numerically. By iterating an RG map, which "course-grains" the description of a many-body system and generates a flow in the parameter space, physical properties of interest can be extracted even for complex models. RG analysis also provides an explanation of physical phenomena such as universality. Many systems exhibit simple RG flows, but more complicated -- even chaotic -- behaviour is also known. Nonetheless, the structure of such RG flows can still be analysed, elucidating the physics of the system, even if specific trajectories may be highly sensitive to the initial point. In contrast, recent work has shown that important physical properties of quantum many-body systems, such as its spectral gap and phase diagram, can be uncomputable. In this work, we show that such undecidable systems exhibit a novel type of RG flow, revealing a qualitatively different and more extreme form of unpredictability than chaotic RG flows. In contrast to chaotic RG flows in which initially close points can diverge exponentially, trajectories under these novel uncomputable RG flows can remain arbitrarily close together for an uncomputable number of iterations, before abruptly diverging to different fixed points that are in separate phases. The structure of such uncomputable RG flows is so complex that it cannot be computed or approximated, even in principle. We give a mathematically rigorous construction of the block-renormalisation-group map for the original undecidable many-body system that appeared in the literature (Cubitt, P\'erez-Garcia, Wolf, Nature 528, 207-211 (2015)). We prove that each step of this RG map is computable, and that it converges to the correct fixed points, yet the resulting RG flow is uncomputable., Comment: 88 pages, 11 figures

In this paper we analyse the birational geometry of O'Grady ten dimensional manifolds, giving a characterisation of Kaehler classes and lagrangian fibrations. Moreover, we study symplectic compactifications of intermediate jacobian fibrations of smooth cubic fourfolds., Comment: Comments are very welcome!

Randomized benchmarking (RB) refers to a collection of protocols that in the past decade have become central methods for characterizing quantum gates. These protocols aim at efficiently estimating the quality of a set of quantum gates in a way that is resistant to state preparation and measurement errors. Over the years many versions have been developed, however, a comprehensive theoretical treatment of RB has been missing. In this work, we develop a rigorous framework of RB general enough to encompass virtually all known protocols as well as novel, more flexible extensions. Overcoming previous limitations on error models and gate sets, this framework allows us, for the first time, to formulate realistic conditions under which we can rigorously guarantee that the output of any RB experiment is well-described by a linear combination of matrix exponential decays. We complement this with a detailed analysis of the fitting problem associated with RB data. We introduce modern signal processing techniques to RB, prove analytical sample complexity bounds, and numerically evaluate performance and limitations. In order to reduce the resource demands of this fitting problem, we introduce novel, scalable post-processing techniques to isolate exponential decays, significantly improving the practical feasibility of a large set of RB protocols. These post-processing techniques overcome shortcomings in efficiency of several previously proposed methods such as character benchmarking and linear-cross entropy benchmarking. Finally, we discuss, in full generality, how and when RB decay rates can be used to infer quality measures like the average fidelity. On the technical side, our work substantially extends the recently developed Fourier-theoretic perspective on RB by making use of the perturbation theory of invariant subspaces, as well as ideas from signal processing., Comment: 61 pages, 7 figures, replaced by final version in PRX Quantum

We prove that any symplectic automorphism of finite order on a manifold of type OG6 acts trivially on the Beauville--Bogomolov--Fujiki lattice and that any birational transformation of finite order acts trivially on its discriminant group. Moreover, we classify all possible invariant and coinvariant sublattices., Comment: Final version, to appear on Kyoto J. Math

We compute the monodromy group of irreducible holomorphic symplectic manifolds of OG10 type, confirming a conjecture of Markman., Comment: 28 pages, comments very welcome. Mild changes with respect to the first version: updated and improved bibliography; cleared some typos; improved some slightly ambiguous statement; corrected a small mistake. Last version is the published: some arguments clarified and expanded

Any technology requires precise benchmarking of its components, and the quantum technologies are no exception. Randomized benchmarking allows for the relatively resource economical estimation of the average gate fidelity of quantum gates from the Clifford group, assuming identical noise levels for all gates, making use of suitable sequences of randomly chosen Clifford gates. In this work, we report significant progress on randomized benchmarking, by showing that it can be done for individual quantum gates outside the Clifford group, even for varying noise levels per quantum gate. This is possible at little overhead of quantum resources, but at the expense of a significant classical computational cost. At the heart of our analysis is a representation-theoretic framework that we develop here which is brought into contact with classical estimation techniques based on bootstrapping and matrix pencils. We demonstrate the functioning of the scheme at hand of benchmarking tensor powers of T-gates. Apart from its practical relevance, we expect this insight to be relevant as it highlights the role of assumptions made on unknown noise processes when characterizing quantum gates at high precision., Comment: 4+13 pages, 4 figures, small changes, references added

Moduli spaces of polarised (with fixed polarisation type) irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on $K3$ surfaces are not connected in general and A. Apostolov counted the number of their components. We extend Apostolov's results to the case of manifolds deformation equivalent to generalised Kummer varieties., Comment: Last version is the published one (title changed according to referee's suggestion)

Random quantum processes play a central role both in the study of fundamental mixing processes in quantum mechanics related to equilibration, thermalisation and fast scrambling by black holes, as well as in quantum process design and quantum information theory. In this work, we present a framework describing the mixing properties of continuous-time unitary evolutions originating from local Hamiltonians having time-fluctuating terms, reflecting a Brownian motion on the unitary group. The induced stochastic time evolution is shown to converge to a unitary design. As a first main result, we present bounds to the mixing time. By developing tools in representation theory, we analytically derive an expression for a local k-th moment operator that is entirely independent of k, giving rise to approximate unitary k-designs and quantum tensor product expanders. As a second main result, we introduce tools for proving bounds on the rate of decoupling from an environment with random quantum processes. By tying the mathematical description closely with the more established one of random quantum circuits, we present a unified picture for analysing local random quantum and classes of Markovian dissipative processes, for which we also discuss applications., Comment: 35 pages, 2 figures, replaced by final version for the Communications in Mathematical Physics