Deep Quantum Neural Networks are Gaussian Process

TL;DR

This paper demonstrates that overparameterized variational quantum circuits behave like Gaussian Processes, providing a new probabilistic framework to analyze quantum neural networks and their kernel properties.

Contribution

It introduces a Gaussian Process perspective for quantum neural networks, linking kernel behavior, finite-width effects, and quantum tangent kernels in a unified framework.

Findings

QNNs behave as Gaussian Processes at wide width or depth.

Finite width effects can be analyzed with a $1/d$ expansion.

Quantum meta-kernels monitor deviations from Gaussian outputs.

Abstract

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 $d$ 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.