Storage capacity and learning capability of quantum neural networks

TL;DR

This paper investigates the storage capacity of quantum neural networks, revealing they can store exponentially more states than classical networks and analyzing the structure and limitations of their state storage capabilities.

Contribution

It provides a theoretical analysis of the storage capacity of QNNs as CPTP maps, including their ability to store exponential states and the structure of these maps, extending to mixed states and input-output relations.

Findings

QNNs can store up to N linearly independent pure states.

QNNs can store an exponential number of states compared to classical networks.

The volume of CPTP maps with M stationary states decreases exponentially with M.

Abstract

We study the storage capacity of quantum neural networks (QNNs) described as completely positive trace preserving (CPTP) maps, which act on an $N$-dimensional Hilbert space. We demonstrate that QNNs can store up to $N$ linearly independent pure states and provide the structure of the corresponding maps. While the storage capacity of a classical Hopfield network scales linearly with the number of neurons, we show that QNNs can store an exponential number of linearly independent states. We estimate, employing the Gardner program, the relative volume of CPTP maps with $M$ stationary states. The volume decreases exponentially with $M$ and shrinks to zero for $M\geq N+1$. We generalize our results to QNNs storing mixed states as well as input-output relations for feed-forward QNNs. Our approach opens the path to relate storage properties of QNNs to the quantum properties of the input-output states. This paper is dedicated to the memory of Peter Wittek.