Absence of Barren Plateaus in Quantum Convolutional Neural Networks

TL;DR

This paper demonstrates that Quantum Convolutional Neural Networks (QCNNs) avoid barren plateau issues, ensuring their trainability and providing a new analytical approach for studying quantum neural network gradients.

Contribution

The paper provides an analytical proof that QCNNs do not exhibit barren plateaus and introduces a novel graph-based method for analyzing Haar-distributed unitaries.

Findings

Gradient variance in QCNNs vanishes no faster than polynomially.

QCNNs are trainable with random initialization.

Numerical simulations confirm analytical predictions.

Abstract

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 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.