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arXiv:2305.09957 · Nature Physics

Quantum neural networks form Gaussian processes

Diego García-Martín, Martín Larocca, Marco Cerezo

Published May 21, 2025• 21 citations
arXivHTMLPDF

Abstract

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 $${\mathcal{O}}\left({1}/{\operatorname{e}^{d}\sqrt{d}}\right)$$ . The connection between classical neural networks and Gaussian processes is a fundamental result in machine learning. It has now been shown that many quantum neural networks converge to Gaussian processes, enabling their use for regression tasks.

DOI
10.1038/s41567-025-02883-z

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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 O(1/e⁡dd){\mathcal{O}}\left({1}/{\operatorname{e}^{d}\sqrt{d}}\right)O(1/edd​) . The connection between classical neural networks and Gaussian processes is a fundamental result in machine learning. It has now been shown that many quantum neural networks converge to Gaussian processes, enabling their use for regression tasks.
Classical neural networks with random initialization famously behave as Gaussian processes in the limit of many neurons, which allows one to completely characterize their training and generalization behavior. No such general understanding exists for quantum neural networks (QNNs), which -- outside of certain special cases -- are known to not behave as Gaussian processes when randomly initialized. We here prove that QNNs and their first two derivatives instead generally form what we call "Wishart processes," where certain algebraic properties of the network determine the hyperparameters of the process. This Wishart process description allows us to, for the first time: give necessary and sufficient conditions for a QNN architecture to have a Gaussian process limit; calculate the full gradient distribution, generalizing previously known barren plateau results; and calculate the local minima distribution of algebraically constrained QNNs. Our unified framework suggests a certain simple operational definition for the "trainability" of a given QNN model using a newly introduced, experimentally accessible quantity we call the "degrees of freedom" of the network architecture.
Abstract We study quantum neural networks where the generated function is the expectation value of the sum of single-qubit observables across all qubits. In (Girardi et al., CMP (2025), it is proven that the probability distributions of such generated functions converge in distribution to a Gaussian process in the limit of infinite width for both untrained networks with randomly initialized parameters and trained networks. In this paper, we provide a quantitative proof of this convergence in terms of the Wasserstein distance of order 1. First, we establish an upper bound on the distance between the probability distribution of the function generated by any untrained network with finite width and the Gaussian process with the same covariance. This proof utilizes Stein’s method to estimate the Wasserstein distance of order 1. Next, we analyze the training dynamics of the network via gradient flow, proving an upper bound on the distance between the probability distribution of the function generated by the trained network and the corresponding Gaussian process. This proof is based on a quantitative upper bound on the maximum variation of a parameter during training. This bound implies that for sufficiently large widths, training occurs in the lazy regime, i.e., each parameter changes only by a small amount. While the convergence result of Girardi et al., CMP (2025) holds at a fixed training time, our upper bounds are uniform in time and hold even as t→∞t \rightarrow \infty t→∞ t → ∞ .
Given a set of input states, we consider the task of predicting the expectation value of a Pauli observable at the output of an unknown quantum evolution, using only a limited number of measurements. Recently, quantum Gaussian process (QGP) regression was introduced for this task across various classes of unitary evolution. Here, we extend the QGP framework beyond unitary dynamics. In particular, we prove convergence of the channel's outputs to a QGP and derive the associated closed-form kernel under a uniform (Lebesgue measure) prior over quantum channels. The kernel's dimensional factor, however, dictates the required observation precision. While manageable when the channel and observable are restricted to small subsystems, exponential suppression precludes learning when the subsystem grows extensively with the system size. Since the Lebesgue prior is overly broad for many applications, we propose an empirical Bayes heuristic that replaces the dimensional factor with a learnable scale parameter while retaining the kernel's state-overlap correlation structure. In numerical simulations of up to 64 qubits, channel QGP regression with the Lebesgue kernel exhibits a strong inductive bias for local channels, enabling faithful extrapolation. For global 64-qubit channels, the rescaled kernel restores learnability, with predictions improving systematically with the shot budget. Results from a noisy quantum computer further demonstrate the robustness of QGP regression under experimental conditions. Beyond regression, we validate QGPs as Bayesian-optimization surrogates for state preparation under noisy XXZ dynamics.
A quantum neural network (QNN) is a parameterized mapping efficiently implementable on near-term Noisy Intermediate-Scale Quantum (NISQ) computers. It can be used for supervised learning when combined with classical gradient-based optimizers. Despite the existing empirical and theoretical investigations, the convergence of QNN training is not fully understood. Inspired by the success of the neural tangent kernels (NTKs) in probing into the dynamics of classical neural networks, a recent line of works proposes to study over-parameterized QNNs by examining a quantum version of tangent kernels. In this work, we study the dynamics of QNNs and show that contrary to popular belief it is qualitatively different from that of any kernel regression: due to the unitarity of quantum operations, there is a non-negligible deviation from the tangent kernel regression derived at the random initialization. As a result of the deviation, we prove the at-most sublinear convergence for QNNs with Pauli measurements, which is beyond the explanatory power of any kernel regression dynamics. We then present the actual dynamics of QNNs in the limit of over-parameterization. The new dynamics capture the change of convergence rate during training and implies that the range of measurements is crucial to the fast QNN convergence.
Quantum neural networks generalize classical artificial neural networks into the quantum domain. They are formulated as parameterized quantum circuits which are optimized by measuring and minimizing a suitably chosen loss function. The core challenge in understanding, implementing and ultimately using quantum neural networks is that they represent many-body systems with an exponentially large Hilbert space, in combination with a large parameter search space. Moreover, noise -- which is inherent to any quantum measurement -- sets practical limits for the estimation of training loss. Here, we study physics-informed large-scale quantum neural networks that are trained through a finite number of noisy loss function measurements. We show that this architecture permits the construction of nontrivial decision boundaries that enable the classification of quantum states through measuring an order parameter. Our approach can directly process quantum data that is output from quantum simulators and computers and is well suited for implementation on current hardware. Moreover, owed to a close link between the neural network dynamics and the evolution of Markovian open many-body quantum systems, one may expect a certain robustness to noise, which is ubiquitous in the current NISQ era.
The overparameterization of variational quantum circuits, as a model of Quantum Neural Networks (QNN), not only improves their trainability but also serves as a method for evaluating the property of a given ansatz by investigating their kernel behavior in this regime. In this study, we shift our perspective from the traditional viewpoint of training in parameter space into function space by employing the Bayesian inference in the Reproducing Kernel Hilbert Space (RKHS). We observe the influence of initializing parameters using random Haar distribution results in the QNN behaving similarly to a Gaussian Process (QNN-GP) at wide width or, empirically, at a deep depth. This outcome aligns with the behaviors observed in classical neural networks under similar circumstances with Gaussian initialization. Moreover, we present a framework to examine the impact of finite width in the closed-form relationship using a $ 1/d$ expansion, where ddd represents the dimension of the circuit's Hilbert space. The deviation from Gaussian output can be monitored by introducing new quantum meta-kernels. Furthermore, we elucidate the relationship between GP and its parameter space equivalent, characterized by the Quantum Neural Tangent Kernels (QNTK). This study offers a systematic way to study QNN behavior in over- and under-parameterized scenarios, based on the perturbation method, and addresses the limitations of tracking the gradient descent methods for higher-order corrections like dQNTK and ddQNTK. Additionally, this probabilistic viewpoint lends itself naturally to accommodating noise within our model.