Quantum-Enhanced In-Context Learning for Geopotential Field Estimation: A Theoretical Framework
M. Hawarey
Abstract
We establish a theoretical framework for in-context learning (ICL) of Earth's gravitational potential field using transformer architectures, with particular emphasis on the sample complexity advantages afforded by quantum gravimetry. The geopotential, expressed as a truncated spherical harmonic expansion of maximum degree N with K = (N+1)² coefficients, defines a function class for which we characterize ICL learnability. Building on the ICL characterization framework of Hawarey (2026) (i.e. the foundational paper), we prove that the geopotential function class is ICL-Easy: it admits an additive sufficient statistic computable by a single attention layer, enabling transformers to match the sample complexity of optimal statistical estimators. Our main contributions are threefold. First, we prove that the minimal sufficient statistic for geopotential ICL is the pair (Gₙ, cₙ) consisting of the Gram matrix and cross-correlation vector, with dimension K² + K, and demonstrate its attention-computability. Second, we derive tight sample complexity bounds showing that ICL achieves nICL = Θ(Kσ²/ε · log(K/δ)) for noise variance σ², target accuracy ε, and failure probability δ, matching empirical risk minimization. Third, we quantify the quantum advantage: for quantum gravimeters operating at the Heisenberg limit with Natoms entangled atoms, the sample complexity reduces by a factor of ρQ = Natoms², yielding up to 10¹²-fold improvement for realistic sensor parameters. We also establish fundamental boundaries: while forward geopotential prediction is ICL-Easy, inverse problems such as density inversion and source localization are ICL-Hard. Source localization with J discrete masses requires identifying combinatorial structure from exponentially many candidates, which we prove satisfies the ICL-Hard structural condition H1 of Hawarey (2026) when J = ω(1), establishing that no polynomial-size transformer can solve it efficiently. This dichotomy—ICL-Easy forward problems versus ICL-Hard inverse problems—reflects the fundamental mathematical structure of potential theory and provides guidance for applying transformer-based methods in geodesy.