[Paper] Exploring the Effect of Basis Rotation on NQS Performance

Source: arXiv - 2512.17893v1

Overview

This paper investigates why neural‑quantum‑state (NQS) models—neural networks that encode many‑body wavefunctions—perform so differently when the underlying quantum basis is rotated. Using an analytically solvable 1‑D Ising chain, the authors show that a simple change of basis can dramatically reshape the geometry of the loss landscape, making the exact solution harder for shallow networks (e.g., RBMs) to find even though the landscape’s “height” (energy) stays the same.

Key Contributions

• Analytical framework: Derives a closed‑form expression for a rotated Ising Hamiltonian, allowing exact tracking of the target wavefunction as the basis angle varies.
• Loss‑landscape invariance: Proves that local basis rotations leave the loss surface (energy as a function of network parameters) unchanged, but move the exact solution to a different region of that surface.
• Information‑geometric diagnostics: Introduces quantum Fisher information (QFI) and Fubini‑Study distance metrics to quantify how far the rotated target state is from typical random initializations.
• Empirical study of shallow NQS: Shows that Restricted Boltzmann Machines trained with the quantum natural gradient often get stuck in saddle‑point or high‑curvature zones, yielding low energy but wrong coefficient distributions.
• Identification of “barrier” phenomena: In the ferromagnetic regime, near‑degenerate eigenstates create narrow, high‑curvature ridges that trap optimization at intermediate fidelities.
• Design recommendation: Highlights the need for architecture‑ and optimizer‑aware design that respects the geometry of the loss landscape rather than treating it as a black box.

Methodology

1. Model system – The authors start from the exactly solvable transverse‑field Ising chain (periodic boundary, spin‑½).
2. Basis rotation – They apply a uniform single‑qubit rotation (R(\theta)=\exp(-i\theta\sigma^y/2)) to every site, which yields a new Hamiltonian (H(\theta)) that is unitarily equivalent to the original.
3. Loss landscape – The variational energy (E(\mathbf{w})=\langle\psi_{\mathbf{w}}|H(\theta)|\psi_{\mathbf{w}}\rangle) is computed for a Restricted Boltzmann Machine (RBM) with parameters (\mathbf{w}). Because the rotation is unitary, the functional form of (E) does not change; only the location of the exact ground‑state parameters (\mathbf{w}^\star(\theta)) moves.
4. Geometric metrics – For each rotation angle they evaluate:
   • Quantum Fisher Information (QFI) of the RBM wavefunction, indicating local curvature of the parameter manifold.
   • Fubini‑Study distance between the exact rotated ground state and the current RBM state, measuring “how far” the optimizer is in Hilbert space.
5. Training protocol – Shallow RBMs (few hidden units) are trained with the quantum natural gradient (QNG), which uses the QFI as a preconditioner, and compared against plain stochastic gradient descent.
6. Diagnostics – After training, they compare energy error, fidelity, and the distribution of wavefunction coefficients to the exact solution.

Results & Findings

Overall, the study confirms that basis rotation does not change the physics but reshapes where the exact solution lives in the neural‑network parameter space, exposing hidden geometric obstacles that shallow NQS struggle to overcome.

Practical Implications

• Model selection: When using NQS for real‑world quantum simulations (e.g., quantum chemistry, condensed‑matter), shallow RBMs may be insufficient if the problem’s natural basis is far from the computational basis. Adding hidden units or switching to more expressive architectures (e.g., deep CNNs, autoregressive models) can alleviate the issue.
• Optimizer design: The quantum natural gradient, while theoretically optimal, can still be misled by high‑curvature saddle regions. Hybrid schemes that combine QNG with curvature‑regularization or adaptive learning‑rate schedules may be more robust.
• Pre‑training & basis engineering: Rotating the input basis to a “more natural” frame (e.g., aligning with dominant interaction terms) can dramatically reduce the geometric distance to the target state, making training faster and more reliable. This suggests a workflow where a cheap classical rotation is applied before launching the NQS optimization.
• Benchmarking standards: Energy‑only metrics are insufficient for evaluating NQS quality. Developers should also report fidelity, coefficient distribution, and information‑geometric diagnostics to catch hidden failures.
• Hardware‑aware implementations: On near‑term quantum‑inspired hardware (e.g., photonic or superconducting RBM chips), the ability to re‑encode the basis efficiently could be a decisive factor for achieving high‑fidelity state preparation.

Limitations & Future Work

• Model scope: The experiments focus on a 1‑D Ising chain with periodic boundaries and uniform rotations. Extending to higher‑dimensional lattices, disordered systems, or non‑uniform rotations may reveal new landscape features.
• Network depth: Only shallow RBMs are examined in depth; while deeper models perform better, the paper does not systematically explore architecture‑specific geometry (e.g., convolutional layers, transformer‑based NQS).
• Optimizer variety: The study contrasts QNG with vanilla SGD. Other second‑order methods (e.g., Kronecker‑factored approximations) or meta‑learning optimizers could be tested against the identified barriers.
• Scalability of diagnostics: Computing QFI and Fubini‑Study distances scales poorly with system size. Future work could develop scalable estimators or surrogate metrics for large‑scale simulations.
• Real‑world applications: Applying the framework to chemically relevant Hamiltonians (e.g., Hubbard or ab‑initio electronic structure) would validate whether the observed basis‑rotation effects persist in more complex settings.

By exposing the hidden geometry that basis rotations induce, this work opens a path toward landscape‑aware NQS design, a crucial step for turning neural quantum states from a research curiosity into a reliable tool for industry‑scale quantum simulations.

Authors

• Sven Benjamin Kožić
• Vinko Zlatić
• Fabio Franchini
• Salvatore Marco Giampaolo

Paper Information

• arXiv ID: 2512.17893v1
• Categories: quant-ph, cs.AI
• Published: December 19, 2025
• PDF: Download PDF