Exploring the Effect of Basis Rotation on NQS Performance

Neural Quantum States (NQS) use neural networks to represent wavefunctions of quantum many-body systems, but their performance depends on the choice of basis, yet the underlying mechanism remains poorly understood. We use a fully solvable one-dimensional Ising model to show that local basis rotations leave the loss landscape unchanged while relocating the exact wavefunction in parameter space, effectively increasing its geometric distance from typical initializations. By sweeping a rotation angle, we compute quantum Fisher information and Fubini-Study distances to quantify how the rotated wavefunction moves within the loss landscape. Shallow architectures (with focus on Restricted Boltzmann Machines (RBMs)) trained with quantum natural gradient are more likely to fall into saddle-point regions depending on the rotation angle: they achieve low energy error but fail to reproduce correct coefficient distributions. In the ferromagnetic case, near-degenerate eigenstates create high-curvature barriers that trap optimization at intermediate fidelities. We introduce a framework based on an analytically solvable rotated Ising model to investigate how relocating the target wavefunction within a fixed loss landscape exposes information-geometric barriers,such as saddle points and high-curvature regions,that hinder shallow NQS optimization, underscoring the need for landscape-aware model design in variational training.