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arXiv:2610.12015

Who can sample forever? Shot noise effects in quantum linear regression

Gabriele Lo Monaco, Salvatore Lorenzo, Alessandro Ferraro, Mauro Paternostro, G. Massimo Palma, Luca Innocenti

quant-phSubmitted Oct 8, 2026• Announced Oct 9, 2026• 0 citations
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Abstract

Quantum fidelity kernels and quantum extreme learning machines share the same basic architecture: a fixed quantum device generates features that are combined through a classically trained linear model. We refer to this broad non-variational setting as quantum linear regression. A defining aspect of these models is that the quantum features are not directly available, but must be estimated from a finite number of shots. We show that finite statistics makes quantum linear regression unavoidably ill-conditioned. Ideal feature vectors are confined to a subspace determined by the dimension of the Hilbert space, whereas finite-shot estimates generically acquire components outside this subspace. Training then attempts to fit directions that disappear in the infinite-statistics limit, making the learned model increasingly sensitive to sampling noise. We analyze this mechanism for quantum extreme learning machines trained on $n_{\rm tr}$ states with $N$ shots per state. Despite the ill-conditioning, predictions remain controlled when the training targets are exact. Finite sampling produces a systematic prediction error that decreases quadratically with $N$, while fluctuations induced by the random training measurements decrease inversely with the total training budget. Thus, unlike in standard linear regression with exactly known features, increasing the dataset size alone cannot remove the error caused by sampling noise. If the training targets are also noisy, due for instance to state-preparation errors, the otherwise hidden directions become active. The test error can then increase with $N$, a behavior that we ascribe to a form of quantum overfitting. Our results expose a fundamental tradeoff between dataset size and per-state statistics in non-variational quantum learning and provide a baseline for resource-aware training and regularization strategies.

Comments
Main: 15 pages, 6 figures. Supplementary: 37 pages, 6 figures

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Tackling output sampling noise due to finite shots of quantum measurement is an unavoidable challenge when extracting information in machine learning with physical systems. A technique called Eigentask Learning was developed recently as a framework for learning with infinite input training data in the presence of output sampling noise. In the work of Eigentask Learning, numerical evidence was presented that extracting low-noise contributions of features can practically improve performance for machine learning tasks, displaying robustness to overfitting and increasing generalization accuracy. However, it remains unsolved to quantitatively characterize generalization errors in situations where the training dataset is finite, while output sampling noise still exists. In this study, we use methodologies from statistical mechanics to calculate the training and generalization errors of a generic quantum machine learning system when the input training dataset and output measurement sampling shots are both finite. Our analytical findings, supported by numerical validation, offer solid justification that Eigentask Learning provides optimal learning in the sense of minimizing generalization errors.
Increasing capacity can induce a machine learning model to generalize worse until it perfectly interpolates the training data, and better once it moves beyond interpolation. This inversion, known as \emph{double descent}, is well understood for independent features. For correlated features, its shape is governed by the spectrum of the feature covariance, but existing analyses typically yield only implicit equations. We first close this gap for classical linear regression, analytically characterizing the entire risk curve through two quantities, namely the condition number of the feature population covariance and a spectral measure of correlation strength. Then, we apply this theory on quantum models, focusing on quantum extreme learning machines (QELMs). We show that three tunable design knobs, \textit{i.e.,} encoding strength, input reuploads, and measurement shots, each regularize the QELM output covariance, and derive conditions under which the interpolation threshold remains stable. Finite measurement shot noise can be leveraged to guarantee this stability. Numerical experiments confirm each prediction, yielding concrete design principles for QELMs that generalize well beyond interpolation.
Abstract Quantum extreme learning machines (QELMs) have emerged as a promising framework for quantum machine learning. Their appeal lies in the rich feature map induced by the dynamics of a quantum substrate—the quantum reservoir—and the efficient post-measurement training via linear regression. Here, we study the expressivity of QELMs by decomposing the prediction of QELMs into a Fourier series. We show that the achievable Fourier frequencies are determined by the data encoding scheme, while Fourier coefficients depend on both the reservoir and the measurement. Notably, the expressivity of QELMs is fundamentally limited by the number of Fourier frequencies and the number of observables, while the complexity of the prediction hinges on the reservoir. As a cautionary note on scalability, we identify four sources that can lead to the exponential concentration of the observables as the system size grows (randomness, hardware noise, entanglement, and global measurements) and show how this can turn QELMs into useless input-agnostic oracles. In particular, our result on the reservoir-induced concentration strongly indicates that quantum reservoirs drawn from a highly random ensemble make QELM models unscalable. Our analysis elucidates the potential and fundamental limitations of QELMs and lays the groundwork for systematically exploring quantum reservoir systems for other machine learning tasks.
Quantum machine learning (QML) models hold promise for leveraging quantum computers in data-driven tasks but their performance on today's noisy quantum hardware remains an open question. In this work, we investigate the impact of shot noise, the statistical noise due to finite sampling in quantum measurements, on the training and performance of quantum neural network (QNN) models. Using a dataset of dielectron collision events from CERN, we train QNNs under varying levels of simulated noise. Three ansatz circuit architectures are explored namely BasicEntanglerLayers (Ansatz-1), SimplifiedTwoDesign (Ansatz-2) and StronglyEntanglingLayers (Ansatz-3). Each model is repeatedly trained for multiple runs at noise levels ranging from zero up to significant noise fractions (0.5), with each training run consisting of 20 epochs of parameter optimization. We evaluate the convergence behavior, final error rates and variability across runs for each noise setting. Our results show a clear degradation of QNN performance as shot noise increases with training loss converging more slowly and to higher values and the model's prediction accuracy deteriorates with larger noise. Notably, the ansatz with fewer parametric gates demonstrates relatively better robustness to noise compared to deeper, more strongly entangling circuits which suffer sharper performance drops under noise. We provide detailed comparisons of the three circuit designs, including training loss curves and statistical summaries of performance at each noise level. These findings shed light on the practical limits of QML models in the Noisy Intermediate-Scale Quantum (NISQ) era and highlight the trade-offs between circuit expressiveness and noise resilience. Finally, we discuss implications for designing noise-aware quantum models and possible strategies to mitigate shot noise effects in quantum learning.
We introduce an end-to-end optimization strategy for quantum machine learning that directly targets performance under finite measurement resources, where learning objectives are defined directly at the level of task performance. The method is applied on a Bayesian quantum metrology task since it provides a natural testbed with known fundamental limits and scaling with system size. The sampling-aware hybrid algorithm achieves a single-shot risk within 1 dB of the -20 dB Bayesian limit using 32 qubits. We extend the Bayesian framework from parameter estimation to global function inference, where the task is to infer a target function of the sensor input drawn from an arbitrary prior, and we demonstrate a clear computational-sensing advantage for direct functional inference over indirect reconstruction. We relate the corresponding Bayesian risk to the Capacity metric and argue that the Resolvable Expressive Capacity provides a natural measure of the space of functions accessible in a single shot. The resulting eigentask analysis identifies noise-robust feature combinations that yield compact estimators with improved accuracy and reduced optimization cost in resource-limited or real-time on-device settings. Going beyond the irreducible statistical uncertainty associated with finite measurement records, we study implementation-induced distortions of the sensor response focusing on representative imperfections arising from readout errors and static coherent gate disorder. We show that when these imperfections are stationary across training and inference, end-to-end optimization can adapt the quantum circuit and classical estimator to the implemented hardware response, preserving high inference performance over a broad range of imperfection strengths.
The double descent phenomenon challenges traditional statistical learning theory by revealing scenarios where larger models do not necessarily lead to reduced performance on unseen data. While this counterintuitive behavior has been observed in a variety of classical machine learning models, particularly modern neural network architectures, it remains elusive within the context of quantum machine learning. In this work, we analytically demonstrate that linear regression models in quantum feature spaces can exhibit double descent behavior by drawing on insights from classical linear regression and random matrix theory. Additionally, our numerical experiments on quantum kernel methods across different real-world datasets and system sizes further confirm the existence of a test error peak, a characteristic feature of double descent. Our findings provide evidence that quantum models can operate in the modern, overparameterized regime without experiencing overfitting, potentially opening pathways to improved learning performance beyond traditional statistical learning theory.