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This paper theoretically analyzes the generalization of Transformer-based neural quantum states (NQS) under in-context learning, establishing an inference-time generalization error bound in terms of MSE. The bound demonstrates that prediction error scales inversely with the number of in-context examples and Transformer depth, with required depth growing only linearly with system size. This framework is extended to full quantum states represented as rank-one density operators, with numerical simulations validating the theoretical findings.
Transformer NQS generalize with prediction error inversely proportional to in-context examples and depth, requiring only linear depth scaling with system size.
Neural quantum states based on modern deep learning architectures have emerged as powerful representations for quantum many-body systems. In particular, Transformer-based neural quantum states provide expressive models capable of capturing long-range correlations, and their empirical generalization performance has recently been demonstrated. However, a theoretical understanding of their generalization behavior remains largely unexplored. In this paper, we develop a theoretical framework to analyze the generalization properties of Transformer-based neural quantum states under in-context learning. We establish a rigorous inference-time generalization error bound in terms of mean squared error (MSE), showing that the pointwise prediction error decreases inversely with both the number of in-context examples and the depth of the Transformer. We further show that the Transformer depth required to achieve this guarantee scales only linearly with the system size--namely, the number of particles in continuous systems or the number of qudits in discrete systems. Building on this result, we extend our analysis to full quantum states formulated as rank-one density operators, and derive MSE-based generalization bounds over both continuous and discrete domains under physical constraints. Finally, numerical simulations corroborate our theoretical analysis.