论文标题
量子神经网络的存储能力和学习能力
Storage capacity and learning capability of quantum neural networks
论文作者
论文摘要
我们研究了描述为完全积极的痕量保存（CPTP）地图的量子神经网络（QNN）的存储能力，该图作用于$ n $维的希尔伯特空间。我们证明QNN最多可以存储多达$ n $的独立纯状状态，并提供相应的地图的结构。尽管经典Hopfield网络的存储容量与神经元的数量线性缩放，但我们表明QNN可以存储指数级的线性独立状态。我们估计，采用Gardner计划，具有$ M $固定状态的CPTP地图的相对体积。 $ m $的体积呈指数下降，并以$ m \ geq n+1 $缩小至零。我们将结果推广到存储混合状态的QNN以及用于前馈QNN的输入输出关系。我们的方法打开了将QNN的存储属性与输入输出状态的量子属性相关联的路径。本文致力于记忆彼得·维特克（Peter Wittek）。
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.
