New preprint Storage capacity and learning capability of quantum neural networks

In this work Maciej Lewenstein, Aikaterini Gratsea, Andreu Riera-Campeny, Albert Aloy, Anna Sanpera and me estimate and in some cases calculate the number of quantum mechanical machines (represented by CPTP maps) fulfilling specific tasks. For more information see https://arxiv.org/abs/2011.06113

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≥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.