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, , 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.

我们研究了量子神经网络 (QNN) 的存储容量，由作用于N维希尔伯特空间的完全正迹保留 (CPTP) 映射描述。我们证明了吸引子 QNN 可以以非平凡的方式存储多达N 个线性独立的纯状态。对于n 个量子位，QNN 可以达到指数存储容量，明显优于标准经典神经网络，其存储容量与神经元数量n成线性关系。我们使用 Gardner 程序估计具有M ⩽ N 个静止状态的 CPTP 映射的相对体积，并表明该体积随M呈指数下降对于M ⩾ N + 1 ，收缩为零。我们将结果推广到存储混合状态的 QNN 以及前馈 QNN 的输入-输出关系。我们的方法开辟了将 QNN 的存储特性与输入-输出状态的量子特征相关联的途径。本文旨在纪念 Peter Wittek。