Hierarchy of discriminative power and complexity in learning quantum ensembles

Distance metrics are central to machine learning, yet distances between ensembles of quantum states remain poorly understood due to fundamental quantum measurement constraints. We introduce a hierarchy of integral probability metrics, termed MMD-$k$, which generalizes the maximum mean discrepancy to quantum ensembles and exhibit a strict trade-off between discriminative power and statistical efficiency as the moment order $k$ increases. For pure-state ensembles of size $N$, estimating MMD-$k$ using experimentally feasible SWAP-test-based estimators requires $Θ(N^{2-2/k})$ samples for constant $k$, and $Θ(N^3)$ samples to achieve full discriminative power at $k = N$. In contrast, the quantum Wasserstein distance attains full discriminative power with $Θ(N^2 \log N)$ samples. These results provide principled guidance for the design of loss functions in quantum machine learning, which we illustrate in the training quantum denoising diffusion probabilistic models.

Transformer-Based Denoising of Mechanical Vibration Signals

Mechanical vibration signal denoising is a pivotal task in various industrial applications, including system health monitoring and failure prediction. This paper introduces a novel deep learning transformer-based architecture specifically tailored for denoising mechanical vibration signals. The model leverages a Multi-Head Attention layer with 8 heads, processing input sequences of length 128, embedded into a 64-dimensional space. The architecture also incorporates Feed-Forward Neural Networks, Layer Normalization, and Residual Connections, resulting in enhanced recognition and extraction of essential features. Through a training process guided by the Mean Squared Error loss function and optimized using the Adam optimizer, the model demonstrates remarkable effectiveness in filtering out noise while preserving critical information related to mechanical vibrations. The specific design and choice of parameters offer a robust method adaptable to the complex nature of mechanical systems, with promising applications in industrial monitoring and maintenance. This work lays the groundwork for future exploration and optimization in the field of mechanical signal analysis and presents a significant step towards advanced and intelligent mechanical system diagnostics.