Entanglement entropy production in Quantum Neural Networks

TL;DR

This study uses tensor network methods to analyze entanglement entropy in quantum neural networks with up to fifty qubits, revealing universal entanglement growth patterns and introducing the concept of entangling speed.

Contribution

It provides the first large-scale quantitative analysis of entanglement in QNNs, demonstrating their approach to Haar-random states and defining a new measure called entangling speed.

Findings

Entanglement in QNNs approaches that of Haar-random states as depth increases.

The entangling speed is a universal characteristic across different QNN architectures.

QNNs generate entanglement at a rate consistent with approximate random unitaries.

Abstract

Quantum Neural Networks (QNN) are considered a candidate for achieving quantum advantage in the Noisy Intermediate Scale Quantum computer (NISQ) era. Several QNN architectures have been proposed and successfully tested on benchmark datasets for machine learning. However, quantitative studies of the QNN-generated entanglement have been investigated only for up to few qubits. Tensor network methods allow to emulate quantum circuits with a large number of qubits in a wide variety of scenarios. Here, we employ matrix product states to characterize recently studied QNN architectures with random parameters up to fifty qubits showing that their entanglement, measured in terms of entanglement entropy between qubits, tends to that of Haar distributed random states as the depth of the QNN is increased. We certify the randomness of the quantum states also by measuring the expressibility of the circuits, as well as using tools from random matrix theory. We show a universal behavior for the rate at which entanglement is created in any given QNN architecture, and consequently introduce a new measure to characterize the entanglement production in QNNs: the entangling speed. Our results characterise the entanglement properties of quantum neural networks, and provides new evidence of the rate at which these approximate random unitaries.