Storage capacity and learning capability of quantum neural networks

We study the storage capacity of quantum neural networks (QNNs), described by completely positive trace preserving (CPTP) maps acting on an N-dimensional Hilbert space. We demonstrate that attractor QNNs can store in a non-trivial manner up to N linearly independent pure states. For n qubits, QNNs can reach an exponential storage capacity, O(2n) , clearly outperforming standard classical neural networks whose storage capacity scales linearly with the number of neurons n. We estimate, employing the Gardner program, the relative volume of CPTP maps with M ⩽ N stationary states and show that this 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 features of the input–output states. This paper is dedicated to the memory of Peter Wittek.