Generalization of Transformer-Based Neural Quantum States via In-Context Learning

In this paper, we establish a collection of covering number bounds for linear function classes under various norm constraints on the inputs and matrices. We then combine these results with existing covering number bounds to derive improved estimates and, based on these estimates, develop generalization error bounds for single-layer Transformers. The resulting generalization bounds improve upon several existing results in the literature and, in particular, are independent of the input sequence length. Moreover, our generalization error bound decays at the rate $O(1/\sqrt{n})$, where $n$ denotes the sample size, thereby improving upon existing bounds that scale as $O((\log n)/\sqrt{n})$. Furthermore, our covering number analysis explicitly incorporates rank constraints on the underlying matrix classes, allowing us to characterize how low-rank structures affect the metric entropy and, consequently, the resulting generalization bounds for Transformer architectures.

Transformers have emerged as powerful architectures for learning solution operators of physical systems. Empirically the prediction error has been observed to decrease when the data size and model size increase, suggesting neural scaling behavior. Yet a theoretical understanding of such scaling laws for transformer-based operator learning remains limited. In this work, we develop a theoretical framework for characterizing the approximation and generalization errors of transformer-based operator learning. Our analysis builds on a local-to-global approximation principle that is naturally aligned with the softmax attention mechanism and yields discretization-invariant output functions. On approximation theory, we derive a universal approximation error of transformer-based operator learning for Hölder-regular operators. On generalization theory, we establish a power scaling law between the prediction error and the training data size. The rate of convergence represented by the scaling exponent explicitly reflects the dimensions of the input and output domains, the regularity of the underlying functions and operators, and crucially, the intrinsic dimension of the input function class. By exploiting this intrinsic low-dimensional structure, our analysis yields a power-law generalization rate for operator learning, in contrast to the logarithmic-type power-law rates appearing in existing analyses of operator learning with feedforward neural networks. Numerical experiments validate the predicted power-law scaling and confirm that the convergence rate varies systematically with the intrinsic dimension of the input function class.

Quantum communication underpins secure information processing and scalable quantum networks. In particular, remote state preparation (RSP) enables efficient quantum state transfer, but accurately estimating target states under complex noise remains challenging. Here, we propose a Transformer-based Quantum State Characterizer (TQSC) model for noisy RSP experiments. Our model reconstructs experimentally prepared pure and mixed photonic polarization states from noisy measurements in complex scattering environments, while its attention patterns provide physically grounded insights into correlations among the measured observables. The method achieves a mean estimator-target fidelity exceeding 99.999% under complex scattering and dynamic Gaussian noise, while its robustness and generalization are further examined using Qiskit-simulated Bloch-ball states. Furthermore, in a practical MNIST image transmission task with held-out states, the decoded bit error rate is reduced from 50.34% to zero after TQSC post-processing. The TQSC model enables accurate tomographic characterization under dynamic noise and provides physically grounded post-hoc insights, holding promise for intelligent quantum information processing applications.