Quantum neural networks form Gaussian processes

Journal Article · Wed May 21 00:00:00 EDT 2025 · Nature Physics

DOI:https://doi.org/10.1038/s41567-025-02883-z· OSTI ID:2567894

García-Martín, Diego ^[1]; Larocca, Martín ^[1]; Cerezo, M. ^[1]

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1. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Classical artificial neural networks initialized from independent and identically distributed priors converge to Gaussian processes in the limit of a large number of neurons per hidden layer. This correspondence plays an important role in the current understanding of the capabilities of neural networks. Here we prove an analogous result for quantum neural networks. We show that the outputs of certain models based on Haar-random unitary or orthogonal quantum neural networks converge to Gaussian processes in the limit of large Hilbert space dimension d. The derivation of this result is more nuanced than in the classical case due to the role played by the input states, the measurement observable and because the entries of unitary matrices are not independent. We show that the efficiency of predicting measurements at the output of a quantum neural network using Gaussian process regression depends on the number of measured qubits. Furthermore, our theorems imply that the concentration of measure phenomenon in Haar-random quantum neural networks is worse than previously thought, because expectation values and gradients concentrate as $$\mathcal{O} (1/e^d \sqrt{d})$$

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE; USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 2567894
• Alternate ID(s): OSTI ID: 2568877
• Report Number(s): LA-UR--23-24867; 1745-2481; 10.1038/s41567-025-02883-z
• Journal Information: Nature Physics, Journal Name: Nature Physics Journal Issue: 7 Vol. 21; ISSN 1745-2473; ISSN 1745-2481
• Publisher: Nature Publishing Group (NPG)
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Barren Plateaus
Gaussian Process
Quantum Neural Networks