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arXiv:2608.08798

Symmetry Constraints Regularize Neural Quantum State Learning

Turbasu Chatterjee, Manas Sajjan, Songbo Xie, Elliott Love, Vinit Singh, Bojko N. Bakalov, Sabre Kais

quant-phSubmitted Aug 9, 2026• Announced Aug 11, 2026• 0 citations
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Abstract

Neural quantum states (NQS) offer highly expressive variational wavefunctions, but their optimization is frequently bottlenecked by redundant parameters and poorly conditioned landscapes. We demonstrate that embedding Hamiltonian symmetries directly into the variational parameterization geometrically regularizes this learning problem. For Boltzmann-family NQS, we enforce symmetries by tying local Pauli-$Z$ generators along physical geometric orbits, analytically collapsing the trainable coefficient space prior to optimization. To quantify the resulting optimization geometry, we introduce a geometric metric built on the Jacobian and Hessian of the optimization landscape. This framework evaluates the fraction of the physically accessible state space that corresponds to high-quality, low-energy solutions. Evaluating our approach on transverse-field Ising (TFIM) and XXZ spin chains shows that symmetry compilation excises the vast majority of parameters while maintaining ground-state accuracy within the resolution of the reported benchmarks. In large TFIM systems, strong spatial constraints compress thousands of parameters down to tens, delivering substantial runtime accelerations. Our geometric diagnostics indicate that symmetry produces a more favorable target-aware geometry by concentrating the reachable state space around low-energy solutions while retaining broad target basins. Together, our results indicate that symmetry compilation concentrates the expressive power of NQS on states relevant to the target problem, thereby reducing model size and training cost without sacrificing accuracy.

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Quantum physics-informed neural networks (QPINNs) have recently emerged as a promising framework for the solution of partial differential equations (PDEs), with several studies reporting improved convergence and accuracy relative to classical physics-informed neural networks (PINNs) at reduced training cost. Motivated by these advances, we introduce geometric quantum physics-informed neural networks (GQPINNs), a symmetry-aware extension of QPINNs in which the geometric structure of the underlying PDE is incorporated directly into the quantum-circuit ansatz. Building on the framework of geometric quantum machine learning, we construct parametrized circuits that encode finite-group and compact Lie-group symmetries as inductive biases through problem-specific equivariant generator sets . Using a twirling-based construction, we derive symmetry-preserving gates that ensure that the model predictions respect the symmetries of the governing equation whenever the boundary and initial data are symmetry compatible. We benchmark GQPINNs against standard QPINNs and symmetry-adapted classical PINN baselines under matched training protocols across a representative set of linear and nonlinear PDEs. Across these benchmarks, GQPINNs achieve improved solution accuracy, as quantified by lower mean absolute error, while requiring substantially fewer trainable parameters. These results identify symmetry-aware quantum-circuit design as an effective route toward improved efficiency and generalization in quantum PDE solvers and provide a systematic framework for incorporating geometric inductive biases into quantum-enhanced scientific machine learning.

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Neural quantum states (NQS) offer highly expressive variational wavefunctions, but their optimization is frequently bottlenecked by redundant parameters and poorly conditioned landscapes. We demonstrate that embedding Hamiltonian symmetries directly into the variational parameterization geometrically regularizes this learning problem. For Boltzmann-family NQS, we enforce symmetries by tying local Pauli-ZZZ generators along physical geometric orbits, analytically collapsing the trainable coefficient space prior to optimization. To quantify the resulting optimization geometry, we introduce a geometric metric built on the Jacobian and Hessian of the optimization landscape. This framework evaluates the fraction of the physically accessible state space that corresponds to high-quality, low-energy solutions. Evaluating our approach on transverse-field Ising (TFIM) and XXZ spin chains shows that symmetry compilation excises the vast majority of parameters while maintaining ground-state accuracy within the resolution of the reported benchmarks. In large TFIM systems, strong spatial constraints compress thousands of parameters down to tens, delivering substantial runtime accelerations. Our geometric diagnostics indicate that symmetry produces a more favorable target-aware geometry by concentrating the reachable state space around low-energy solutions while retaining broad target basins. Together, our results indicate that symmetry compilation concentrates the expressive power of NQS on states relevant to the target problem, thereby reducing model size and training cost without sacrificing accuracy.
Neural Quantum States (NQS) are powerful variational representations of quantum many-body wavefunctions, yet their performance depends sensitively on the chosen basis. Using an exactly solvable one-dimensional Ising model, we show that local basis rotations leave the minimization landscape unchanged while relocating the exact ground state in parameter space. This provides a controlled framework to disentangle representational limitations from optimization-induced trainability effects. This geometric displacement, quantified through information-geometric measures, can steer optimization of shallow architectures toward saddle points and high-curvature regions. As a result, low energy errors may coexist with an incorrect wavefunction structure. By comparing energy and infidelity optimization within the same variational architectures, we show that optimization failure can persist even when the rotated target state remains representable. Our results identify a geometric mechanism contributing to basis dependence in NQS and motivate landscape-aware variational design.
Neural quantum states (NQS) provide a flexible and scalable framework for approximating quantum many-body wavefunctions. Among NQS parameterizations, autoregressive models are especially attractive because they enable exact, independent sampling from the Born distribution, avoiding the autocorrelation and mixing issues of Markov chain methods. Yet their optimization remains comparatively underexplored: Adam is a scalable method but ignores function space geometry, while stochastic reconfiguration is principled but costly and numerically fragile in large models. To address this gap, we show that variational energy minimization can be viewed as an advantage policy-gradient problem over the Born distribution, motivating trust-region optimization for NQS training. We introduce Proximal Wavefunction Optimization (PWO), a principled trust-region algorithm that clips probability-ratio changes in the amplitude channel and phase increments in the phase channel. PWO avoids explicit matrix inversion, reuses samples across multiple updates, and combines the scalability of first-order optimization with theoretical guarantees. Across Ising and frustrated J1J_1J1​-J2J_2J2​ one- and two-dimensional spin systems, PWO improves stability and wall-clock convergence over Adam, minSR, and SPRING. Finally, we fine-tune a 1.51.51.5B-parameter RWKV-7 model, demonstrating NQS optimization at a scale over three orders of magnitude beyond prior work.
Neural Quantum States (NQS) provide a powerful neural network-based variational framework for representing many-body wave functions and solving for ground states. Recurrent Neural Networks (RNNs) are particularly promising owing to their relatively low computational cost and their autoregressive property, which enables perfect sampling. Recently, RNNs have been reported to be unstable under curvature-based optimizers such as the minimum-step stochastic reconfiguration (minSR) method. In this paper, we address this perceived limitation and show that minSR can be stabilized through simple regularization techniques, enabling robust training of RNN-based NQS with only a few samples. Our approach outperforms the Adam optimizer on the one-dimensional transverse-field Ising model and the one-dimensional cluster state, and provides competitive results on the two-dimensional Heisenberg and J1−J2J_1-J_2J1​−J2​ models. This work offers a promising pathway for using modern optimization techniques with autoregressive NQS to address open questions in quantum simulation.
Neural quantum states (NQS) have proven highly effective in representing quantum many-body wavefunctions, but their loss landscape remains poorly understood and debated. Here, we demonstrate that the NQS loss landscape is more benign and similar to conventional deep learning than previously thought, exhibiting mode connectivity: independently trained NQS are connected by paths in parameter space with essentially no energy barrier. To construct these paths, we develop GeoNEB, a path optimizer integrating efficient stochastic reconfiguration with the nudged elastic band method for constructing minimum energy paths. For the strongly interacting six-electron quantum dot modeled by a 1.61.61.6M-parameter Psiformer, we find two independent minima with expected energy barrier ∼10−5\sim10^{-5}∼10−5 times smaller than the system's overall energy scale and ∼10−3\sim10^{-3}∼10−3 times smaller than the linear path's barrier. The path respects physical symmetry in addition to achieving low energy, with the angular momentum remaining well quantized throughout. Our work is the first to construct optimized paths between independently trained NQS, and it suggests that the NQS loss landscape may not be as pathological as once feared.
Neural quantum states (NQS) have emerged as a powerful framework for simulating non-equilibrium dynamics in strongly correlated quantum systems, offering scalable variational representations of highly entangled states. Yet, accurate NQS simulations have been found to be surprisingly challenging in some physical regimes of limited complexity. Here, we address paradigmatic quench dynamics of a one-dimensional quantum Ising model as a controlled benchmark. Through supervised state reconstruction we establish substantially tighter empirical upper bounds on the required parameter count than previous estimates, ruling out representational limitations as the key obstruction. Instead, we uncover a geometric inflation of small deviations as a hitherto overlooked challenge for accurate solutions of the infinitesimal time-dependent variational principle (TDVP): the dynamical rotation of the kernel of the quantum geometric tensor (QGT) can suddenly lend physical significance to previously irrelevant parameter deviations. The stability of matrix product state solutions of the same TDVP suggests that the non-linearity of the neural network ansatz is the origin of the sensitivity. These results identify QGT-null-space rotation as a geometric diagnostic of sensitive NQS dynamics and as a concrete target for improving TDVP algorithms.
Quantum physics-informed neural networks (QPINNs) have recently emerged as a promising framework for the solution of partial differential equations (PDEs), with several studies reporting improved convergence and accuracy relative to classical physics-informed neural networks (PINNs) at reduced training cost. Motivated by these advances, we introduce geometric quantum physics-informed neural networks (GQPINNs), a symmetry-aware extension of QPINNs in which the geometric structure of the underlying PDE is incorporated directly into the quantum-circuit ansatz. Building on the framework of geometric quantum machine learning, we construct parametrized circuits that encode finite-group and compact Lie-group symmetries as inductive biases through problem-specific equivariant generator sets . Using a twirling-based construction, we derive symmetry-preserving gates that ensure that the model predictions respect the symmetries of the governing equation whenever the boundary and initial data are symmetry compatible. We benchmark GQPINNs against standard QPINNs and symmetry-adapted classical PINN baselines under matched training protocols across a representative set of linear and nonlinear PDEs. Across these benchmarks, GQPINNs achieve improved solution accuracy, as quantified by lower mean absolute error, while requiring substantially fewer trainable parameters. These results identify symmetry-aware quantum-circuit design as an effective route toward improved efficiency and generalization in quantum PDE solvers and provide a systematic framework for incorporating geometric inductive biases into quantum-enhanced scientific machine learning.