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arXiv:2408.11901

A Unified Theory of Quantum Neural Network Loss Landscapes

Eric R. Anschuetz

Published Aug 21, 2024• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2408.11901

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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.
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.
Quantum neural networks (QNNs) have generated excitement around the possibility of efficiently analyzing quantum data.But this excitement has been tempered by the existence of exponentially vanishing gradients, known as barren plateau landscapes, for many QNN architectures.Recently, quantum convolutional neural networks (QCNNs) have been proposed, involving a sequence of convolutional and pooling layers that reduce the number of qubits while preserving information about relevant data features.In this work, we rigorously analyze the gradient scaling for the parameters in the QCNN architecture.We find that the variance of the gradient vanishes no faster than polynomially, implying that QCNNs do not exhibit barren plateaus.This result provides an analytical guarantee for the trainability of randomly initialized QCNNs, which highlights QCNNs as being trainable under random initialization unlike many other QNN architectures.To derive our results, we introduce a novel graph-based method to analyze expectation values over Haar-distributed unitaries, which will likely be useful in other contexts.Finally, we perform numerical simulations to verify our analytical results.
One of the most important properties of neural networks is the clustering of local minima of the loss function near the global minimum, enabling efficient training. Though generative models implemented on quantum computers are known to be more expressive than their traditional counterparts, it has empirically been observed that these models experience a transition in the quality of their local minima. Namely, below some critical number of parameters, all local minima are far from the global minimum in function value; above this critical parameter count, all local minima are good approximators of the global minimum. Furthermore, for a certain class of quantum generative models, this transition has empirically been observed to occur at parameter counts exponentially large in the problem size, meaning practical training of these models is out of reach. Here, we give the first proof of this transition in trainability, specializing to this latter class of quantum generative model. We use techniques inspired by those used to study the loss landscapes of classical neural networks. We also verify that our analytic results hold experimentally even at modest model sizes.
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 → ∞ .
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.
Quantum neural networks (QNNs) have been a promising framework in pursuing near-term quantum advantage in various fields, where many applications can be viewed as learning a quantum state that encodes useful data. As a quantum analog of probability distribution learning, quantum state learning is theoretically and practically essential in quantum machine learning. In this paper, we develop a no-go theorem for learning an unknown quantum state with QNNs even starting from a high-fidelity initial state. We prove that when the loss value is lower than a critical threshold, the probability of avoiding local minima vanishes exponentially with the qubit count, while only grows polynomially with the circuit depth. The curvature of local minima is concentrated to the quantum Fisher information times a loss-dependent constant, which characterizes the sensibility of the output state with respect to parameters in QNNs. These results hold for any circuit structures, initialization strategies, and work for both fixed ansatzes and adaptive methods. Extensive numerical simulations are performed to validate our theoretical results. Our findings place generic limits on good initial guesses and adaptive methods for improving the learnability and scalability of QNNs, and deepen the understanding of prior information's role in QNNs.