Exploring the Effect of Basis Rotation on NQS Performance

TL;DR

The paper investigates how basis choices influence Neural Quantum States by relocating the exact ground state within a fixed loss landscape of a solvable 1D Ising model. It uses shallow NQS (RBMs and small FFNNs) optimized with quantum natural gradient and analyzes the state-space displacement through quantum Fisher information and the Fubini–Study distance. Key findings show that basis rotations can push the target into information-geometric saddle points or high-curvature regions, causing low-fidelity coefficient structures despite small energy errors, with distinct behavior in ferromagnetic and antiferromagnetic regimes. The work argues for landscape-aware ansatz design, possibly deeper architectures and hybrid loss strategies, and positions the rotated Ising model as a valuable diagnostic testbed for variational quantum optimization.

Abstract

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.

Figures (7)

• Figure 1: UMAP projection of several NQS trajectories indicating different landscapes according to (\ref{['eq:energy']}) and (\ref{['eq:infidelity']}) respectively. Color denotes the infidelity with respect to the ground state of ferromagentic Ising model with $N=5, h=0.5$
• Figure 2: Toy example of the smooth map defined in \ref{['eq:smooth_map']} that shows how the curvature and the saddle point occurring in the loss function space influences the parameter manifold $\Theta$, causing another saddle point in the input space.
• Figure 3: Comparison of Fubini Study Distance $\gamma$ and Loss $\mathcal{L}$ from NQS trained with infidelity minimization up to initial state $\lvert \psi\rangle = 1/\sqrt{2^{N}}\sum_{s\in\{\pm1\}^N} \lvert s\rangle$ and then to the target rotated state ground state $|\varphi(\phi)\rangle$ for $N=5, h=0.5$ with $J=-1$ (top) and $J=+1$ (bottom) as function of rotation angle $\phi$.
• Figure 4: Discussion in Sec. \ref{['sec:considerations']} can help us unify the understanding in variational optimization of different algorithms. In this case we compare the performance of Lanczos algorithm Lanczos_1950, DMRG and RBM with $\alpha=4$ starting from the same initial statevector and the ferromagnetic Ising model $N=5, h=-0.5, \phi=\pi/3$. Left top plot shows energy $E$ during iterations $\tau$ while bottom plot paints the colormap for infidelity $1-\mathcal{F}$. All three algorithms encounter the same saddle point region. The right plot compares the corresponding statevector trajectories using UMAP projection.
• Figure 5: Relative Error versus angle of rotation $\phi$ for logRBM ($\alpha=1$) with SR for $\tau=5000$ iterations for rotated $H(\phi) = U_{\phi} H {U_{\phi}}^{\dagger}$\ref{['eq:Hamiltonian']}