Accurate battery health prognosis using State of Health (SOH) estimation is essential for the reliability of multiscale battery energy storage, yet existing methods are limited in generalizability across diverse battery chemistries and operating conditions. The inability of standard neural networks to capture the complex, high-dimensional physics of battery degradation is a major contributor to these limitations. To address this, a physics-informed neural network with the Quantum Feature Mapping(QFM) technique (QPINN) is proposed. QPINN projects raw battery sensor data into a high-dimensional Hilbert space, creating a highly expressive feature set that effectively captures subtle, non-linear degradation patterns using Nyström method. These quantum-enhanced features are then processed by a physics-informed network that enforces physical constraints. The proposed method achieves an average SOH estimation accuracy of 99.46% across different datasets, substantially outperforming state-of-the-art baselines, with reductions in MAPE and RMSE of up to 65% and 62%, respectively. This method was validated on a large-scale, multi-chemistry dataset of $\mathbf{3 1 0, 7 0 5}$ samples from 387 cells, and further showed notable adaptability in cross-validation settings, successfully transferring from one chemistry to another without relying on target-domain SOH labels.