Conformal prediction can be used to construct prediction sets that cover the true outcome with a desired probability, but can sometimes lead to large prediction sets that are costly in practice. The most useful outcome is a singleton prediction-an unambiguous decision-yet existing efficiency-oriented methods primarily optimize average set size. Motivated by this, we propose a new nonconformity score that aims to minimize the probability of producing non-singleton sets. Starting from a non-convex constrained optimization problem as a motivation, we provide a geometric reformulation and associated algorithm for computing the nonconformity score and associated split conformal prediction sets in O(K) time for K-class problems. Using this score in split conformal prediction leads to our proposed Singleton-Optimized Conformal Prediction (SOCOP) method. We evaluate our method in experiments on image classification and LLM multiple-choice question-answering, comparing with standard nonconformity scores such as the (negative) label probability estimates and their cumulative distribution function; both of which are motivated by optimizing length. The results show that SOCOP increases singleton frequency (sometimes by over 20%) compared to the above scores, with minimal impact on average set size.

While virtual reality (VR) excels at simulating physical environments, its effectiveness for training complex interpersonal skills is limited by a lack of psychologically plausible virtual humans. This gap is particularly critical in medical education, where communication is a core clinical competency. This paper introduces a framework that integrates large language models (LLMs) into immersive VR to create medically coherent virtual patients with distinct, consistent personalities, based on a modular architecture that decouples personality from clinical data. We evaluated the system in a mixed-methods, within-subjects study with licensed physicians conducting simulated consultations. Results suggest that the approach is feasible and perceived as a rewarding and effective training enhancement. Our analysis highlights key design principles, including a "realism-verbosity paradox" and the importance of challenges being perceived as clinically authentic to support learning.

Accurate compensation of brain shift is critical for maintaining the reliability of neuronavigation during neurosurgery. While keypoint-based registration methods offer robustness to large deformations and topological changes, they typically rely on simple geometric interpolators that ignore tissue biomechanics to create dense displacement fields. In this work, we propose a novel deep learning framework that estimates dense, physically plausible brain deformations from sparse matched keypoints. We first generate a large dataset of synthetic brain deformations using biomechanical simulations. Then, a residual 3D U-Net is trained to refine standard interpolation estimates into biomechanically guided deformations. Experiments on a large set of simulated displacement fields demonstrate that our method significantly outperforms classical interpolators, reducing by half the mean square error while introducing negligible computational overhead at inference time. Code available at: \href{https://github.com/tiago-assis/Deep-Biomechanical-Interpolator}{https://github.com/tiago-assis/Deep-Biomechanical-Interpolator}.

Understanding and reasoning about code semantics is essential for enhancing code LLMs' abilities to solve real-world software engineering (SE) tasks. Although several code reasoning benchmarks exist, most rely on synthetic datasets or educational coding problems and focus on coarse-grained reasoning tasks such as input/output prediction, limiting their effectiveness in evaluating LLMs in practical SE contexts. To bridge this gap, we propose CodeSense, the first benchmark that makes available a spectrum of fine-grained code reasoning tasks concerned with the software engineering of real-world code. We collected Python, C and Java software projects from real-world repositories. We executed tests from these repositories, collected their execution traces, and constructed a ground truth dataset for fine-grained semantic reasoning tasks. We then performed comprehensive evaluations on state-of-the-art LLMs. Our results show a clear performance gap for the models to handle fine-grained reasoning tasks. Although prompting techniques such as chain-of-thought and in-context learning helped, the lack of code semantics in LLMs fundamentally limits models' capabilities of code reasoning. Besides dataset, benchmark and evaluation, our work produced an execution tracing framework and tool set that make it easy to collect ground truth for fine-grained SE reasoning tasks, offering a strong basis for future benchmark construction and model post training. Our code and data are located at https://codesense-bench.github.io/.

Learning expressive kernels while retaining tractable inference remains a central challenge in scaling Gaussian processes (GPs) to large and complex datasets. We propose a scalable GP regressor based on deep basis kernels (DBKs). Our DBK is constructed from a small set of neural-network-parameterized basis functions with an explicit low-rank structure. This formulation immediately enables linear-complexity inference with respect to the number of samples, possibly without inducing points. DBKs provide a unifying perspective that recovers sparse deep kernel learning and Gaussian Bayesian last-layer methods as special cases. We further identify that naively maximizing the marginal likelihood can lead to oversimplified uncertainty and rank-deficient solutions. To address this, we introduce a mini-batch stochastic objective that directly targets the predictive distribution with decoupled regularization. Empirically, DBKs show advantages in predictive accuracy, uncertainty quantification, and computational efficiency across a range of large-scale regression benchmarks.

Designing metal-organic frameworks (MOFs) with novel chemistries is a longstanding challenge due to their large combinatorial space and complex 3D arrangements of the building blocks. While recent deep generative models have enabled scalable MOF generation, they assume (1) a fixed set of building blocks and (2) known local 3D coordinates of building blocks. However, this limits their ability to (1) design novel MOFs and (2) generate the structure using novel building blocks. We propose a two-stage MOF generation framework that overcomes these limitations by modeling both chemical and geometric degrees of freedom. First, we train an SMILES-based autoregressive model to generate metal and organic building blocks, paired with a cheminformatics toolkit for 3D structure initialization. Second, we introduce a flow matching model that predicts translations, rotations, and torsional angles to assemble the blocks into valid 3D frameworks. Our experiments demonstrate improved reconstruction accuracy, the generation of valid, novel, and unique MOFs, and the ability to create novel building blocks. Our code is available at https://github.com/nayoung10/MOFFlow-2.

Current Deep Neural Network (DNN) verifiers are typically designed to prioritize scalability over reliability. Reliability can be reinforced through the generation of proofs that are checkable by trusted, external proof checkers. To date, only a handful of verifiers support proof production; and these rely on verifier-specific formats, and balance between scalability, proof detail, and the trustworthiness of their proof checker. In this tool paper, we introduce PICID, a DNN verifier that produces proofs in the standard Alethe format for SMT solving, checkable by multiple existing checkers. PICID implements a parallel CDCL(T) architecture that integrates a state-of-the-art, proof-producing SAT solver with the Marabou DNN verifier. Furthermore, PICID leverages UNSAT proofs to derive conflict clauses. Our evaluation shows that PICID generates valid proofs in the vast majority of cases and significantly outperforms existing tools that produce comparable proofs.

Smart homes equipped with ambient sensors offer a transformative approach to continuous health monitoring and assisted living. Traditional research in this domain primarily focuses on Human Activity Recognition (HAR), which relies on mapping sensor data to a closed set of predefined activity labels. However, the fixed granularity of these labels often constrains their practical utility, failing to capture the subtle, household-specific nuances essential, for example, for tracking individual health over time. To address this, we propose DISCOVER, a framework for discovering and annotating Patterns of Daily Living (PDL) - fine-grained, recurring sequences of sensor events that emerge directly from a resident's unique routines. DISCOVER utilizes a self-supervised feature extraction and representation-aware clustering pipeline, supported by a custom visualization interface that enables experts to interpret and label discovered patterns with minimal effort. Our evaluation across multiple smart-home environments demonstrates that DISCOVER identifies cohesive behavioral clusters with high inter-rater agreement while achieving classification performance comparable to fully-supervised baselines using only 0.01% of the labels. Beyond reducing annotation overhead, DISCOVER establishes a foundation for longitudinal analysis. By grounding behavior in a resident's specific environment rather than rigid semantic categories, our framework facilitates the observation of within-person habitual drift. This capability positions the system as a potential tool for identifying subtle behavioral indicators associated with early-stage cognitive decline in future longitudinal studies.

Recently, over-the-air federated learning (FL) has attracted significant attention for its ability to enhance communication efficiency. However, the performance of over-the-air FL is often constrained by device selection strategies and signal aggregation errors. In particular, neglecting straggler devices in FL can lead to a decline in the fairness of model updates and amplify the global model's bias toward certain devices' data, ultimately impacting the overall system performance. To address this issue, we propose a joint device selection and transmit power optimization framework that ensures the appropriate participation of straggler devices, maintains efficient training performance, and guarantees timely updates. First, we conduct a theoretical analysis to quantify the convergence upper bound of over-the-air FL under age-of-information (AoI)-based device selection. Our analysis further reveals that both the number of selected devices and the signal aggregation errors significantly influence the convergence upper bound. To minimize the expected weighted sum peak age of information, we calculate device priorities for each communication round using Lyapunov optimization and select the highest-priority devices via a greedy algorithm. Then, we formulate and solve a transmit power and normalizing factor optimization problem for selected devices to minimize the time-average mean squared error (MSE). Experimental results demonstrate that our proposed method offers two significant advantages: (1) it reduces MSE and improves model performance compared to baseline methods, and (2) it strikes a balance between fairness and training efficiency while maintaining satisfactory timeliness, ensuring stable model performance.

Radio deployments and spectrum planning benefit from path loss predictions. Obstructions along a communications link are often considered implicitly or through derived metrics such as representative clutter height or total obstruction depth. In this paper, we propose a path-specific path loss prediction method that uses convolutional neural networks to automatically perform feature extraction from 2-D obstruction height maps. Our methods result in low prediction error in a variety of environments without requiring derived metrics.

Physics-Informed Neural Networks (PINNs) solve forward PDEs by minimizing residual losses from the governing equations with initial and boundary conditions, but they often struggle with discontinuities such as shocks. In contrast, finite volume methods (FVM) handle discontinuities by enforcing integral conservation, which admits weak solutions. Motivated by this, we propose a Coupled Integral PINN (CI-PINN) that augments a standard PINN with an auxiliary network for integral potentials and coupled integral constraints. This improves robustness near shocks while avoiding meshing and the numerical flux integration/reconstruction used in classical schemes. We validate CI-PINN on forward benchmarks including Burgers, Buckley--Leverett, the Euler system, and the Shallow-Water equations.

We study efficient mechanism design for allocating multiple heterogeneous objects. The aim is to maximize the residual surplus, the total value generated from an allocation minus the costs of screening. We discover a robust trend indicating that no-screening mechanisms, such as serial dictatorship with exogenous priority order, tend to perform better as the variety of goods increases. We analyze the underlying reasons by characterizing asymptotically efficient mechanisms in a stylized environment. We also apply an automated mechanism design approach to numerically derive efficient mechanisms and validate the trend in general environments. Building on these implications, we propose the register-invite-book system (RIB) as an efficient system for scheduling vaccinations against pandemic diseases.

High-resolution climate simulations are valuable for understanding climate change impacts. This has motivated use of regional convection-permitting climate models (CPMs), but these are very computationally expensive. We present a convection-permitting model generative emulator (CPMGEM), to skilfully emulate precipitation simulations by a 2.2km-resolution regional CPM at much lower cost. This utilises a generative machine learning approach, a diffusion model. It takes inputs at the 60km resolution of the driving global climate model and downscales these to 8.8km, with daily-mean time resolution, capturing the effect of convective processes represented in the CPM at these scales. The emulator is trained on simulations over England and Wales from the United Kingdom Climate Projections Local product, covering years between 1980 and 2080 following a high emissions scenario. The output precipitation has a similar spatial structure and intensity distribution as in the CPM simulations. The emulator is stochastic, which improves the realism of samples. We include some evidence about the emulator's skill for extreme events with return times up to ~100 years. We demonstrate successful transfer from a "perfect model" training setting to application using GCM variable inputs. It captures the main features of the simulated 21st century climate change, but exhibits some error in the magnitude. We also show that the method can be useful in situations with limited amounts of high-resolution data. Potential applications include producing high-resolution precipitation predictions for large-ensemble climate simulations and producing output based on different GCMs and climate change scenarios to better sample uncertainty.

We study the problem of learning a partially observed matrix under the low rank assumption in the presence of fully observed side information that depends linearly on the true underlying matrix. This problem consists of an important generalization of the Matrix Completion problem, a central problem in Statistics, Operations Research and Machine Learning, that arises in applications such as recommendation systems, signal processing, system identification and image denoising. We formalize this problem as an optimization problem with an objective that balances the strength of the fit of the reconstruction to the observed entries with the ability of the reconstruction to be predictive of the side information. We derive a mixed-projection reformulation of the resulting optimization problem and present a strong semidefinite cone relaxation. We design an efficient, scalable alternating direction method of multipliers algorithm that produces high quality feasible solutions to the problem of interest. Our numerical results demonstrate that in the small rank regime ({\color{black}$k \leq 10$}), our algorithm outputs solutions that achieve on average {\color{black}$2.3\%$} lower objective value and {\color{black}$41\%$} lower $\ell_2$ reconstruction error than the solutions returned by the best performing benchmark method on synthetic data. The runtime of our algorithm is competitive with and often superior to that of the benchmark methods. Our algorithm is able to solve problems with $n = 10000$ rows and $m = 10000$ columns in less than a minute. On large scale real world data, our algorithm produces solutions that achieve $67\%$ lower out of sample error than benchmark methods in $97\%$ less execution time.

Semantic code search, retrieving code that matches a given natural language query, is an important task to improve productivity in software engineering. Existing code search datasets face limitations: they rely on human annotators who assess code primarily through semantic understanding rather than functional verification, leading to potential inaccuracies and scalability issues. Additionally, current evaluation metrics often overlook the multi-choice nature of code search. This paper introduces CoSQA+, pairing high-quality queries from CoSQA with multiple suitable codes. We develop an automated pipeline featuring multiple model-based candidate selections and the novel test-driven agent annotation system. Among a single Large Language Model (LLM) annotator and Python expert annotators (without test-based verification), agents leverage test-based verification and achieve the highest accuracy of 93.9%. Through extensive experiments, CoSQA+ has demonstrated superior quality over CoSQA. Models trained on CoSQA+ exhibit improved performance. We publicly release both CoSQA+_all, which contains 412,080 agent-annotated pairs, and CoSQA+_verified, which contains 1,000 human-verified pairs, at https://github.com/DeepSoftwareAnalytics/CoSQA_Plus.

Several recent papers have proposed increasing the expressive power of graph neural networks by exploiting subgraphs or other topological structures. In parallel, researchers have investigated higher order permutation equivariant networks. In this paper we tie these two threads together by providing a general framework for higher order permutation equivariant message passing in subgraph neural networks. In this paper we introduce a new type of mathematical object called $P$-tensors, which provide a simple way to define the most general form of permutation equivariant message passing in both the above two categories of networks. We show that the P-Tensors paradigm can achieve state-of-the-art performance on benchmark molecular datasets.

The performance of automatic speaker verification (ASV) systems could be degraded by voice spoofing attacks. Most existing works aimed to develop standalone spoofing countermeasure (CM) systems. Relatively little work targeted at developing an integrated spoofing aware speaker verification (SASV) system. In the recent SASV challenge, the organizers encourage the development of such integration by releasing official protocols and baselines. In this paper, we build a probabilistic framework for fusing the ASV and CM subsystem scores. We further propose fusion strategies for direct inference and fine-tuning to predict the SASV score based on the framework. Surprisingly, these strategies significantly improve the SASV equal error rate (EER) from 19.31% of the baseline to 1.53% on the official evaluation trials of the SASV challenge. We verify the effectiveness of our proposed components through ablation studies and provide insights with score distribution analysis.

In this paper, we present UR-AIR system submission to the logical access (LA) and the speech deepfake (DF) tracks of the ASVspoof 2021 Challenge. The LA and DF tasks focus on synthetic speech detection (SSD), i.e. detecting text-to-speech and voice conversion as spoofing attacks. Different from previous ASVspoof challenges, the LA task this year presents codec and transmission channel variability, while the new task DF presents general audio compression. Built upon our previous research work on improving the robustness of the SSD systems to channel effects, we propose a channel-robust synthetic speech detection system for the challenge. To mitigate the channel variability issue, we use an acoustic simulator to apply transmission codec, compression codec, and convolutional impulse responses to augmenting the original datasets. For the neural network backbone, we propose to use Emphasized Channel Attention, Propagation and Aggregation Time Delay Neural Networks (ECAPA-TDNN) as our primary model. We also incorporate one-class learning with channel-robust training strategies to further learn a channel-invariant speech representation. Our submission achieved EER 20.33% in the DF task; EER 5.46% and min-tDCF 0.3094 in the LA task.

Spoofing countermeasure (CM) systems are critical in speaker verification; they aim to discern spoofing attacks from bona fide speech trials. In practice, however, acoustic condition variability in speech utterances may significantly degrade the performance of CM systems. In this paper, we conduct a cross-dataset study on several state-of-the-art CM systems and observe significant performance degradation compared with their single-dataset performance. Observing differences of average magnitude spectra of bona fide utterances across the datasets, we hypothesize that channel mismatch among these datasets is one important reason. We then verify it by demonstrating a similar degradation of CM systems trained on original but evaluated on channel-shifted data. Finally, we propose several channel robust strategies (data augmentation, multi-task learning, adversarial learning) for CM systems, and observe a significant performance improvement on cross-dataset experiments.

Precontact manifolds extend contact geometry by weakening the maximal non-integrability condition of the defining $1$-form. We clarify the geometric foundations of this structure by studying general pairs of a $1$-form and a $2$-form under mild regularity conditions. We characterize them through their class, analyse the role of distinguished vector fields, such as Reeb or Liouville fields, and study other associated geometrical objects. Precontact structures are then treated as the special case of pairs formed by a nowhere-vanishing $1$-form and its exterior derivative. We also define Hamiltonian dynamics on precontact manifolds. Several examples are presented to illustrate the theory.

We introduce a propagation-based approach to thermal state simulation by adapting Pauli and Majorana propagation to imaginary-time evolution in the Schrödinger picture. Our key observation is that high-temperature states can be sparse in the Pauli or Majorana bases, approaching the identity at infinite temperature. By formulating imaginary-time evolution directly in these operator bases and evolving from the maximally mixed state, we access a continuum of temperatures where the state remains efficiently representable. We provide analytic guarantees for small-coefficient truncation and Pauli-weight (Majorana-length) truncation strategies by quantifying the error growth and the impact of backflow. Large-scale numerics on the 1D J1-J2 model (energies) and the triangular-lattice Hubbard model (static correlations) validate efficiency at high temperatures.

The classical Erdős-Kac theorem states that for $n$ chosen uniformly at random from $1, \dots, N$, the random variable $(ω(n) - \log\log N)/\sqrt{\log\log N}$ converges in distribution to the standard Gaussian as $N$ tends to infinity. Banks and Shparlinski showed that this Gaussian convergence holds for any Beatty sequence $\lfloorαn + β\rfloor$ in place of $n$. Continuing in this spirit, Crnčević, Hernández, Rizk, Sereesuchart and Tao considered the joint distribution of $ω(n)$ and $ω(\lfloorαn\rfloor)$, which they showed to be asymptotically independent for irrational values of $α$. Generalising both results, we show that for any positive integer $k$, real numbers $α_1, \dots, α_k > 0$ and $β_1, \dots, β_k$, where $α_i/α_j$ is irrational for $i\neq j$, the joint distribution of $(ω(\lfloorα_in + β_i\rfloor) - \log\log N)/\sqrt{\log\log N}$ converges to the $k$-dimensional standard Gaussian. We next discuss quantitative bounds on the rate of convergence which do not depend on the values taken by the relevant parameters. Banks and Shparlinski remarked that such quantitative bounds may be given for a single Beatty sequence $\lfloorαn + β\rfloor$ under Diophantine type assumptions on $α$. We show that such assumptions are in fact unnecessary. Specifically, for any real numbers $α> 0, β$, we show that the Kolmogorov distance between the random variable $(ω(\lfloorαn + β\rfloor) - \log\log N)/\sqrt{\log\log N}$ and the standard Gaussian is bounded above by $O(\log\log\log N/\sqrt{\log\log N})$ as $N$ tends to infinity. On the other hand, we show that universal quantitative bounds of this kind do not exist for higher-degree generalised polynomials or for the joint convergence for multiple Beatty sequences.

This study explores how EU integration, globalisation, and geopolitical disruptions have influenced scientific collaboration among European countries at different stages of EU membership. Specifically, it distinguishes between the EU-14, the EU-13, that joined the EU in 2004 or later, and EU candidate countries. Using Scopus article, the study analyses Relative Intensity of Collaboration (RIC) among EU member state, candidate countries and China, Latin America, the UK, the USA and Russia. Findings indicate increasing integration within European groups and with global partners, yet persistent hierarchical structures remain. EU-14 countries form the core of the network, exhibiting stable and cohesive collaboration, including with the UK despite Brexit. EU-13 countries occupy an intermediate position, showing moderate collaboration with EU-14 but stronger collaboration within their own group, with EU candidate countries and Russia. EU candidate countries demonstrate even weaker integration with EU-14, focusing on intra-group ties and links with EU-13 and Russia. RIC peaks in 2012 and 2018 for EU-13 and EU candidate countries correspond to Horizon 2020 and Horizon Europe cycles, highlighting the role of EU Framework Programmes. Collaboration with Russia increased following 2014 and only marginally declined after 2022. For EU-14, it exceeds collaboration with the USA. Collaboration with China remains limited due to network and cultural constraints, with similar intensity across all three groups. Overall, funding and policy initiatives are critical for stable international collaboration.

We investigate the hardware requirements for quantum teleportation in an intercity-scale network topology consisting of two metropolitan-scale networks connected via a long-distance backbone link. Specifically, we identify the minimal improvements required beyond the state-of-the-art to achieve an end-to-end expected teleportation fidelity of $2/3$, which represents the classical limit. To this end, we formulate the hardware requirements computation as optimisation problems, where the hardware parameters representing the underlying device capabilities serve as decision variables. Assuming a simplified noise model, we derive closed-form analytical expressions for the teleportation fidelity and rate when the network is realised using heterogeneous quantum hardware, including a quantum repeater chain with a memory cut-off. Our derivations are based on events defined by the order statistics of link generation durations in both the metropolitan networks and the backbone, and the resulting expressions are validated through simulations on the NetSquid platform. The analytical expressions facilitate efficient exploration of the optimisation parameter space without resorting to computationally intensive simulations. We then apply this framework to a representative realisation in which the metropolitan nodes are based on trapped-ion processors and the backbone is composed of ensemble-based quantum memories. Our results suggest that teleportation across metropolitan distances is already achievable with state-of-the-art hardware when the data qubit is prepared after end-to-end entanglement has already been established, whereas extending teleportation to intercity scales requires additional, though plausibly achievable, improvements in hardware performance.

We present a simple explicit family $\mathcal{B}$ of $910$ $6$-subsets of $[60]=\{1,\dots,60\}$ such that every $6$-subset $S\subset[60]$ intersects at least one block $B\in\mathcal{B}$ in at least three elements, i.e.\ $|S\cap B|\ge 3$. Equivalently, $\mathcal{B}$ is a covering (dominating set) of the Johnson graph $J(60,6)$ with covering radius $3$ in the Johnson metric. The construction is purely combinatorial, based on a fixed split of $[60]$ into two halves, a pairing of each half, and a pigeonhole argument. We also record a crude counting lower bound and a straightforward generalization to $[2m]$ (with $m$ even).

We use admissible covers to characterize irreducible stable curves that are $(d,h)$-elliptic, that is, that are limits of smooth curves admiting finite maps of degree-$d$ to smooth curves of genus $h\geq 1$.

Low Earth orbit (LEO) satellite systems experience significant Doppler effects due to high mobility. While Doppler shifts can be largely compensated, residual frequency uncertainty induces a structured form of channel uncertainty that can limit achievable rates. We model this effect using a block-fading channel of the form $ \mathbf{H} = \mathbf{F} + s \mathbf{G} $, where $s$ is an unknown scalar random parameter. We first study this model in a general $N\times N$ MIMO setting. For this channel, we derive achievable rate lower bounds based on explicit transmission schemes and capacity upper bounds using a duality approach. We study Gaussian signaling and propose a practical superposition scheme with subspace alignment (SN) and successive interference cancellation, where a coarse-layer stream serves as an implicit pilot for decoding refined-layer data. We characterize asymptotic capacity in the near-coherent and high-SNR regimes, and show via Doppler-OFDM simulations that the proposed SN scheme achieves near-optimal rates with low complexity.

We study conformal tractor bundles from an extrinsic viewpoint, relating them to codimension two spacelike immersions into Lorentzian manifolds. We show that, at least locally, every Riemannian conformal structure admits a natural realization of its normal conformal tractor bundle as the pullback of the tangent bundle of a suitably constructed Lorentzian ambient space. Finally, we reformulate the classical equations characterizing parallel sections of the normal conformal tractor bundle in this extrinsic setting, showing that they can be expressed entirely in terms of the geometry of the associated spacelike immersion. This extrinsic perspective provides additional geometric insight into parallel standard tractors and conformal holonomy.

Quantum mechanics provides cryptographic primitives whose security is grounded in hardness assumptions independent of those underlying classical cryptography. However, existing proposals require low-noise quantum communication and long-lived quantum memory, capabilities which remain challenging to realize in practice. In this work, we introduce a quantum digital signature scheme that operates with only classical communication, using the classical shadows of states produced by random circuits as public keys. We provide theoretical and numerical evidence supporting the conjectured hardness of learning the private key (the circuit) from the public key (the shadow). A key technical ingredient enabling our scheme is an improved state-certification primitive that achieves higher noise tolerance and lower sample complexity than prior methods. We realize this certification by designing a high-rate error-detecting code tailored to our random-circuit ensemble and experimentally generating shadows for 32-qubit states using circuits with $\geq 80$ logical ($\geq 582$ physical) two-qubit gates, attaining 0.90 $\pm$ 0.01 fidelity. With increased number of measurement samples, our hardware-demonstrated primitives realize a proof-of-principle quantum digital signature, demonstrating the near-term feasibility of our scheme.

This paper addresses openness, density and structural stability conditions of one-parameter families of 3D piecewise smooth vector fields (PSVFs) defined around typical singularities. Our treatment is local and the switching set, $M$, is a $2D$ surface embedded in $\mathbb{R}^3$. In short, we analyze the robustness and normal forms of certain codimension one singularities that occur in PSVFs. The main machinery used in this paper involves the theory of contact between a vector field and $M$, Bifurcation Theory and the Topology of Manifolds. Our main result states robust mathematical statements resembling the classical Kupka-Smale Theorem in the sense that we establish the openness and density of a large class of PSVFs presenting generic and quasi-generic singularities. Due to the lack of uniqueness of certain solutions associated with PSVFs, we employ Filippov's theory as the basis of our approach throughout the paper.