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arXiv:2609.23334

Stochastic Reconfiguration as Statistical Filtering for Overparameterized Neural Quantum States

Tak Hur

quant-phcs.LGSubmitted Sep 20, 2026• Announced Sep 22, 2026• 0 citations
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Abstract

Stochastic reconfiguration (SR) is the standard optimizer for neural quantum states (NQS), but modern NQS often have far more parameters than Monte Carlo samples. We show that in this regime the diagonal shift is more than a numerical stabilizer. It acts as a statistical filter for finite-sample generalization. At a fixed wave function, SR is ridge regression from tangent features to the centered local energy. Its residual is the expressivity gap, the part of imaginary-time evolution outside the current tangent space. This gap is orthogonal to the tangent space in population, but finite batches make it act as noise that SR can overfit. The shift therefore balances shrinkage of useful update directions against variance from fitting sampled residuals. Exact diagnostics on a $4\times4$ Heisenberg graph separate two effects of overparameterization. Larger tangent spaces help when they reduce the expressivity gap, but they can hurt when they overfit a fixed gap. In a $100$-site transverse-field Ising family trained with a foundation NQS, validation risk is U-shaped in the shift while variance decreases, matching the noisy-ridge model. This view leads to multi-shift SR (MS-SR), which averages independent ridge solves at data-adaptive shifts to form a richer, lower-variance spectral filter. Checkpoint-local experiments show that MS-SR lowers validation risk and update variance relative to the fixed-shift SR baseline. We further compare MS-SR and SR in paired online training continuations, with independent endpoint energy evaluations and a separate update-cost benchmark.

Comments
25 pages, 7 figures

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Stochastic reconfiguration (SR) is the standard optimizer for neural quantum states (NQS), but modern NQS often have far more parameters than Monte Carlo samples. We show that in this regime the diagonal shift is more than a numerical stabilizer. It acts as a statistical filter for finite-sample generalization. At a fixed wave function, SR is ridge regression from tangent features to the centered local energy. Its residual is the expressivity gap, the part of imaginary-time evolution outside the current tangent space. This gap is orthogonal to the tangent space in population, but finite batches make it act as noise that SR can overfit. The shift therefore balances shrinkage of useful update directions against variance from fitting sampled residuals. Exact diagnostics on a 4×44\times44×4 Heisenberg graph separate two effects of overparameterization. Larger tangent spaces help when they reduce the expressivity gap, but they can hurt when they overfit a fixed gap. In a 100100100-site transverse-field Ising family trained with a foundation NQS, validation risk is U-shaped in the shift while variance decreases, matching the noisy-ridge model. This view leads to multi-shift SR (MS-SR), which averages independent ridge solves at data-adaptive shifts to form a richer, lower-variance spectral filter. Checkpoint-local experiments show that MS-SR lowers validation risk and update variance relative to the fixed-shift SR baseline. We further compare MS-SR and SR in paired online training continuations, with independent endpoint energy evaluations and a separate update-cost benchmark.
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Variational Monte Carlo (VMC) combined with expressive neural network wavefunctions has become a powerful route to high-accuracy ground-state calculations, yet its practical success hinges on efficient and stable wavefunction optimization. While stochastic reconfiguration (SR) provides a geometry-aware preconditioner motivated by imaginary-time evolution, its Kaczmarz-inspired variant, subsampled projected-increment natural gradient descent (SPRING), achieves state-of-the-art empirical performance. However, the effectiveness of SPRING is highly sensitive to the choice of a momentum-like parameter μμμ. The original sensitivity of μμμ and the instability observed at μ=1μ=1μ=1, have remained unclear. In this work, we clarify the distinct mechanisms governing the regimes μ<1μ<1μ<1 and μ=1μ=1μ=1. We establish convergence guarantees for 0≤μ<10\leμ<10≤μ<1 under mild assumptions, and construct counterexamples showing that μ=1μ=1μ=1 can induce divergence via uncontrolled growth along kernel-related directions when the step-size is not summable. Motivated by these theoretical insights and numerical observations, we further propose \textit{Principal Range Informed MomEntum SR} (PRIME-SR), a tuning-free momentum-adaptive SR method based on effective spectral dimension and subspace overlap. PRIME-SR achieves performance comparable to optimally tuned SPRING while significantly improving robustness in VMC optimization.
The representation of a quantum wave function as a neural network quantum state (NQS) provides a powerful variational ansatz for finding the ground states of many-body quantum systems. Nevertheless, due to the complex variational landscape, traditional methods often employ the computation of quantum geometric tensor, consequently complicating optimization techniques. Contributing to efforts aiming to formulate alternative methods, we introduce an approach that bypasses the computation of the metric tensor and instead relies exclusively on first-order gradient descent with Euclidean metric. This allows for the application of larger neural networks and the use of more standard optimization methods from other machine learning domains. Our approach leverages the principle of imaginary time evolution by constructing a target wave function derived from the Schrödinger equation, and then training the neural network to approximate this target. We make this method adaptive and stable by determining the optimal time step and keeping the target fixed until the energy of the NQS decreases. We demonstrate the benefits of our scheme via numerical experiments with 2D J_1-J_2 J 1 − J 2 , Heisenberg model, which showcase enhanced stability and energy accuracy in comparison to direct energy loss minimization. Importantly, our approach displays competitiveness with the well-established density matrix renormalization group method and NQS optimization with stochastic reconfiguration.
This thesis studies the intersection of quantum computing and artificial intelligence in two directions: quantum methods for machine learning and machine learning methods for quantum systems. For quantum machine learning, Neural Quantum Embedding learns data representations that increase the trace distance between embedded class ensembles, lowering an embedding-dependent bound on empirical risk and improving classification on noisy quantum hardware. A training objective based on the Hilbert-Schmidt inner product extends this approach to deterministic quantum computation with one qubit (DQC1) and is demonstrated on an NMR quantum processor. A margin-based generalization analysis then connects quantum neural network performance to quantum state discrimination. In the studied benchmarks, margin distributions predict generalization more reliably than parameter-count metrics. For quantum systems, a Mamba-based neural decoder for surface codes matches a reproduced Transformer baseline in memory experiments while reducing inference-cost scaling from quartic to quadratic in code distance. Under an explicit decoder-induced-noise model, it achieves lower logical error rates and a higher effective threshold. For neural quantum states, stochastic reconfiguration is interpreted as tangent-space ridge regression, with its diagonal shift controlling the bias-variance trade-off under finite Monte Carlo sampling. Multi-shift stochastic reconfiguration reduces checkpoint-local validation residuals and update variance relative to fixed-shift SR, at additional computational cost. Together, these contributions show how learned representations, statistical control, and hardware constraints shape the exchange between quantum computing and machine learning.
Due to the strong correlations present in quantum systems, classical machine learning algorithms like stochastic gradient descent are often insufficient for the training of neural network quantum states (NQSs). These difficulties can be overcome by using physically inspired learning algorithm, the most prominent of which is the stochastic reconfiguration (SR) which mimics imaginary time evolution. Here we explore an alternative algorithms for the optimization of complex valued NQSs based on the linear method (LM), and present the explicit formulation in terms of complex valued parameters. Beyond the theoretical formulation, we present numerical evidence that the LM can be used successfully for the optimization of complex valued NQSs, to our knowledge for the first time. We compare the LM to the state-of-the-art SR algorithm and find that the LM requires up to an order of magnitude fewer iterations for convergence, albeit at a higher cost per epoch. We further demonstrate that the LM becomes the more efficient training algorithm whenever the cost of sampling is high. This advantage, however, comes at the price of a larger variance.