⚛️ quantum physics

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

This paper demonstrates that finite-shot sampling noise fundamentally limits quantum linear regression by rendering the problem ill-conditioned, causing prediction errors to persist or even increase with larger datasets when targets are noisy due to a form of quantum overfitting.

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

Published 2026-10-09

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CC BY 4.0

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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the rapidly evolving field of quantum machine learning, researchers are trying to teach computers to recognize patterns using the strange rules of quantum physics. A common approach involves a two-step process: first, a fixed quantum device takes an input and transforms it into a set of numbers called features, which describe the input in a new way. Second, a standard computer program learns to combine these features to make a prediction, such as identifying a specific chemical property or a type of material. This method is attractive because the quantum part does not need to be trained, avoiding some of the most difficult problems that plague other quantum algorithms. However, there is a fundamental catch: the quantum device cannot simply output these feature numbers directly. Instead, it must measure the quantum state many times to estimate the probabilities of different outcomes. This process is like trying to guess the exact proportion of red and blue marbles in a jar by pulling out a handful; the more marbles you pull, the closer your estimate gets to the truth, but with a limited number of pulls, your estimate will always contain some random error.

A team of physicists has now investigated how this unavoidable measurement error, known as shot noise, affects the ability of these quantum models to learn. They focused on a specific setup where the quantum device generates features from a collection of training data, and a classical computer tries to find the best mathematical relationship between those features and the desired answer. The researchers discovered that the very act of estimating these features with a finite number of measurements creates a hidden trap. In an ideal world with infinite measurements, the features would lie on a flat, lower-dimensional surface within a vast mathematical space. But in the real world, where measurements are limited, the random noise pushes the estimated features off this surface and into the empty space around it. When the computer tries to learn from this noisy data, it attempts to fit a line to these stray points, effectively learning to predict directions that do not actually exist in the underlying physics.

The study reveals that this creates a specific type of error that behaves differently than errors in classical computer science. In standard learning, if you have a noisy dataset, adding more examples usually helps you get closer to the correct answer. Here, the researchers found that simply adding more training examples does not fix the problem if the number of measurements taken for each example remains low. Instead, the error settles at a stubborn floor determined by the precision of the individual measurements. The only way to lower this floor is to increase the number of measurements taken for every single training example. If the researchers tried to learn with too few measurements per example, the model would become increasingly sensitive to the noise, and adding more data would actually stop improving the results.

The situation becomes even more complex when the answers the model is trying to learn are themselves slightly wrong, perhaps due to imperfections in how the quantum states were prepared. In this scenario, the model begins to learn the noise itself. The researchers observed a phenomenon they call quantum overfitting, where increasing the precision of the training data actually makes the model's predictions worse. This happens because the model starts to latch onto the random directions created by the measurement noise, and as the training data becomes more precise, it amplifies the errors in the target answers. The study shows that there is a delicate trade-off: one must balance the number of training examples against the number of measurements taken for each one.

Through detailed mathematical analysis and computer simulations, the team mapped out exactly how these errors grow and shrink. They showed that when the target answers are perfect, the error decreases rapidly at first as measurements increase, but then slows down, dominated by a systematic bias that cannot be removed by adding more data. When the target answers are noisy, the error can eventually start to rise again if the training measurements become too precise without a corresponding increase in the precision of the test measurements. This work provides a clear guide for how to allocate resources in quantum experiments, showing that a larger dataset is not always better if the quality of the individual measurements is not also improved. The findings suggest that future quantum learning systems must be designed with a careful awareness of these statistical limits, treating the number of measurements and the number of examples as two sides of the same coin rather than independent variables.

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

Problem Statement
Quantum linear regression (QLR) encompasses non-variational quantum machine learning architectures, such as Quantum Extreme Learning Machines (QELMs) and quantum fidelity kernels, where a fixed quantum device maps input data into a quantum feature space, and a classical linear model is trained on the resulting expectation values. A fundamental distinction between QLR and classical linear regression is that quantum features are not directly accessible; they must be estimated from a finite number of measurement shots ( $N$ ). This introduces intrinsic sampling noise (shot noise) into the feature matrix.

The paper addresses a critical gap in understanding how this finite-statistics noise affects learning performance. While classical regression with noisy features (errors-in-variables models) is known to introduce bias, the specific structural consequences of shot noise on the rank and conditioning of quantum feature matrices have not been systematically characterized. Specifically, the authors investigate how the interplay between the number of training states ( $n_{tr}$ ) and the number of shots per state ( $N$ ) determines the mean-squared error (MSE), challenging the intuition that increasing the dataset size alone is sufficient to minimize error.

Methodology
The authors analyze the QLR setting where training targets are initially assumed exact, and noise arises solely from the estimation of input probabilities. They model the empirical feature matrix $\hat{P}_N$ as a perturbation of the ideal, rank-deficient matrix $P$ (where $\text{rank}(P) \le d^2$ , with $d$ being the Hilbert space dimension).

Rank Discontinuity Analysis: The core theoretical obstacle identified is that while the ideal matrix $P$ is rank-deficient (confined to a subspace of dimension $d^2$ ), the empirical matrix $\hat{P}_N$ becomes full rank with high probability due to shot noise. This rank increase creates a discontinuity in the Moore-Penrose pseudoinverse $\hat{P}_N^+$ , leading to divergent terms as $N \to \infty$ if not handled correctly.
Perturbative Expansion: The authors develop a tailored perturbative approach for the pseudoinverse in the regime where $n_{tr}, n_{out} > d^2$ . By decomposing the noise matrix into blocks relative to the support and cokernel of $P$ , they isolate the divergent components. They show that while the pseudoinverse itself diverges, physical predictions (involving exact targets and test probabilities) remain finite because the divergent blocks are annihilated by the projection onto the signal subspace.
Bias-Variance Decomposition: The study derives an explicit decomposition of the infinite-test-statistics MSE into training bias and training variance. They employ two complementary approaches:
- A homogeneous Gaussian noise model to analytically compute moments of the Schur complement and derive scaling laws.
- A rigorous multinomial noise analysis that avoids Gaussian approximations, utilizing concentration inequalities and high-probability bounds to control the remainder terms for finite $N$ and large $n_{tr}$ .
Extension to Noisy Targets: The framework is extended to scenarios where training targets are also noisy (e.g., due to state preparation errors), analyzing how this interacts with the rank-increasing shot noise.

Key Contributions and Results

Intrinsic Ill-Conditioning: The paper establishes that finite-shot estimation renders quantum linear regression unavoidably ill-conditioned. The empirical feature matrix acquires "spurious directions" outside the ideal subspace, carrying singular values at the noise scale $N^{-1/2}$ .
Resource Scaling Laws: The authors derive the scaling of the MSE with respect to $N$ $N$ and $n_{tr}$ $n_{t r}$ , identifying two distinct regimes:
- Low Statistics ( $N \lesssim n_{tr}$ ): The MSE is dominated by a systematic bias scaling as $N^{-2}$ . This bias arises from the rank increase and the associated ill-conditioning. Crucially, this bias is insensitive to the number of training states $n_{tr}$ .
- High Statistics ( $N \gtrsim n_{tr}$ ): The MSE is dominated by variance (fluctuations) scaling as $(N n_{tr})^{-1}$ .
The Trade-off: Unlike classical regression with exact features, increasing the dataset size ( $n_{tr}$ ) alone cannot remove the error caused by shot noise once the variance is suppressed. The error saturates at a floor determined by the per-state shot budget ( $N^{-2}$ ). To reduce error further, one must increase $N$ alongside $n_{tr}$ .
Quantum Overfitting: When training targets are noisy, the previously harmless "spurious directions" (created by shot noise) couple to the label noise. This leads to a term in the MSE scaling as $N/M$ (where $M$ is test shots). Consequently, increasing training statistics $N$ without increasing test statistics $M$ can increase the test error. The authors identify this as a form of quantum overfitting, where refining the training features amplifies the noise in the labels.
Explicit Coefficients: The paper provides explicit formulas for the bias and variance coefficients in terms of the POVM, training ensemble, and target observable, validated against numerical simulations across various regimes and dimensions.

Significance and Claims
The paper claims to expose a fundamental trade-off between dataset size and per-state measurement statistics in non-variational quantum learning. It argues that familiar intuitions from classical learning must be revised to account for the geometry of quantum state space, specifically the constraint that ideal quantum features live in a subspace of dimension $d^2$ , while their estimates do not.

The authors emphasize that their results provide a baseline for resource-aware training and regularization strategies. They assert that performance guarantees in Quantum Machine Learning (QML) must explicitly account for finite-shot effects, treating the total experimental budget as a joint allocation problem between the number of training states and the number of shots per state, rather than prioritizing dataset size in isolation. The work highlights that finite statistics can qualitatively alter conclusions drawn from idealized, infinite-sample analyses.

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