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 a $N$-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, $\mathcal O(2^{n})$, 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 $M\leq N$ stationary states and show that this 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 features of the input-output states.  This work is dedicated to the memory of Peter Wittek.