Recent advancements highlight the pivotal role of quantum machine learning (QML) [4, 13] in processing quantum data derived from quantum systems [14]. A fundamental task in QML is generating quantum data by learning the underlying distribution, essential for understanding quantum systems, synthesizing new samples, and advancing applications in quantum chemistry and materials science. However, extending classical generative approaches to quantum data presents significant challenges, as quantum distributions exhibit superposition, entanglement, and non-locality that classical models struggle to replicate efficiently. Quantum generative models such as quantum generative adversarial networks [24, 42] and quantum variational autoencoders [20, 38] can be used to prepare a fixed single quantum state [21, 28, 37], but are inefficient for generating ensembles of quantum states [3] due to the need for training deep parameterized quantum circuits (PQCs). The quantum denoising diffusion probabilistic model [40] offers a promising approach that employs intermediate training steps to smoothly interpolate between the target distribution and noise, thereby enabling efficient training.

AI models like GPT-5 are an increasingly valuable tool for scientists, but many remain unaware of the capabilities of frontier AI. We present a collection of short case studies in which GPT-5 produced new, concrete steps in ongoing research across mathematics, physics, astronomy, computer science, biology, and materials science. In these examples, the authors highlight how AI accelerated their work, and where it fell short; where expert time was saved, and where human input was still key. We document the interactions of the human authors with GPT-5, as guiding examples of fruitful collaboration with AI. Of note, this paper includes four new results in mathematics (carefully verified by the human authors), underscoring how GPT-5 can help human mathematicians settle previously unsolved problems. These contributions are modest in scope but profound in implication, given the rate at which frontier AI is progressing.

Clustering on NISQ hardware is constrained by data loading and limited qubits. We present \textbf{qc-kmeans}, a hybrid compressive $k$-means that summarizes a dataset with a constant-size Fourier-feature sketch and selects centroids by solving small per-group QUBOs with shallow QAOA circuits. The QFF sketch estimator is unbiased with mean-squared error $O(\varepsilon^2)$ for $B,S=Θ(\varepsilon^{-2})$, and the peak-qubit requirement $q_{\text{peak}}=\max\{D,\lceil \log_2 B\rceil + 1\}$ does not scale with the number of samples. A refinement step with elitist retention ensures non-increasing surrogate cost. In Qiskit Aer simulations (depth $p{=}1$), the method ran with $\le 9$ qubits on low-dimensional synthetic benchmarks and achieved competitive sum-of-squared errors relative to quantum baselines; runtimes are not directly comparable. On nine real datasets (up to $4.3\times 10^5$ points), the pipeline maintained constant peak-qubit usage in simulation. Under IBM noise models, accuracy was similar to the idealized setting. Overall, qc-kmeans offers a NISQ-oriented formulation with shallow, bounded-width circuits and competitive clustering quality in simulation.

Quantized neural network training optimizes a discrete, non-differentiable objective. The straight-through estimator (STE) enables backpropagation through surrogate gradients and is widely used. While previous studies have primarily focused on the properties of surrogate gradients and their convergence, the influence of quantization hyperparameters, such as bit width and quantization range, on learning dynamics remains largely unexplored. We theoretically show that in the high-dimensional limit, STE dynamics converge to a deterministic ordinary differential equation. This reveals that STE training exhibits a plateau followed by a sharp drop in generalization error, with plateau length depending on the quantization range. A fixed-point analysis quantifies the asymptotic deviation from the unquantized linear model. We also extend analytical techniques for stochastic gradient descent to nonlinear transformations of weights and inputs.

Federated learning (FL) can dramatically speed up reinforcement learning by distributing exploration and training across multiple agents. It can guarantee an optimal convergence rate that scales linearly in the number of agents, i.e., a rate of $\tilde{O}(1/(NT)),$ where $T$ is the iteration index and $N$ is the number of agents. However, when the training samples arise from a Markov chain, existing results on TD learning achieving this rate require the algorithm to depend on unknown problem parameters. We close this gap by proposing a two-timescale Federated Temporal Difference (FTD) learning with Polyak-Ruppert averaging. Our method provably attains the optimal $\tilde{O}(1/NT)$ rate in both average-reward and discounted settings--offering a parameter-free FTD approach for Markovian data. Although our results are novel even in the single-agent setting, they apply to the more realistic and challenging scenario of FL with heterogeneous environments.

Learning-enabled planning and control algorithms are increasingly popular, but they often lack rigorous guarantees of performance or safety. We introduce an algorithm for computing underapproximate backward reachable sets of nonlinear discrete time neural feedback loops. We then use the backward reachable sets to check goal-reaching properties. Our algorithm is based on overapproximating the system dynamics function to enable computation of underapproximate backward reachable sets through solutions of mixed-integer linear programs. We rigorously analyze the soundness of our algorithm and demonstrate it on a numerical example. Our work expands the class of properties that can be verified for learning-enabled systems.

Identifying low-dimensional latent structures within high-dimensional data has long been a central topic in the machine learning community, driven by the need for data compression, storage, transmission, and deeper data understanding. Traditional methods, such as principal component analysis (PCA) and autoencoders (AE), operate in an unsupervised manner, ignoring label information even when it is available. In this work, we introduce a unified method capable of learning latent spaces in both unsupervised and supervised settings. We formulate the problem as a nonlinear multiple-response regression within an index model context. By applying the generalized Stein's lemma, the latent space can be estimated without knowing the nonlinear link functions. Our method can be viewed as a nonlinear generalization of PCA. Moreover, unlike AE and other neural network methods that operate as "black boxes", our approach not only offers better interpretability but also reduces computational complexity while providing strong theoretical guarantees. Comprehensive numerical experiments and real data analyses demonstrate the superior performance of our method.

We introduce a locally differentially private (LDP) algorithm for online federated learning that employs temporally correlated noise to improve utility while preserving privacy. To address challenges posed by the correlated noise and local updates with streaming non-IID data, we develop a perturbed iterate analysis that controls the impact of the noise on the utility. Moreover, we demonstrate how the drift errors from local updates can be effectively managed for several classes of nonconvex loss functions. Subject to an $(\epsilon,\delta)$-LDP budget, we establish a dynamic regret bound that quantifies the impact of key parameters and the intensity of changes in the dynamic environment on the learning performance. Numerical experiments confirm the efficacy of the proposed algorithm.

Breakthroughs in machine learning (ML) and advances in quantum computing (QC) drive the interdisciplinary field of quantum machine learning to new levels. However, due to the susceptibility of ML models to adversarial attacks, practical use raises safety-critical concerns. Existing Randomized Smoothing (RS) certification methods for classical machine learning models are computationally intensive. In this paper, we propose the combination of QC and the concept of discrete randomized smoothing to speed up the stochastic certification of ML models for discrete data. We show how to encode all the perturbations of the input binary data in superposition and use Quantum Amplitude Estimation (QAE) to obtain a quadratic reduction in the number of calls to the model that are required compared to traditional randomized smoothing techniques. In addition, we propose a new binary threat model to allow for an extensive evaluation of our approach on images, graphs, and text.

We consider the problem of Bayesian regression with trustworthy uncertainty quantification. We define that the uncertainty quantification is trustworthy if the ground truth can be captured by intervals dependent on the predictive distributions with a pre-specified probability. Furthermore, we propose, Trust-Bayes, a novel optimization framework for Bayesian meta learning which is cognizant of trustworthy uncertainty quantification without explicit assumptions on the prior model/distribution of the functions. We characterize the lower bounds of the probabilities of the ground truth being captured by the specified intervals and analyze the sample complexity with respect to the feasible probability for trustworthy uncertainty quantification. Monte Carlo simulation of a case study using Gaussian process regression is conducted for verification and comparison with the Meta-prior algorithm.