Quantum machine learning models based on parametrized quantum circuits, also called quantum neural networks (QNNs), are considered to be among the most promising candidates for applications on near-term quantum devices. Here we explore the expressivity and inductive bias of QNNs by exploiting an exact mapping from QNNs with inputs $x$ to classical perceptrons acting on $x \otimes x$ (generalised to complex inputs). The simplicity of the perceptron architecture allows us to provide clear examples of the shortcomings of current QNN models, and the many barriers they face to becoming useful general-purpose learning algorithms. For example, a QNN with amplitude encoding cannot express the Boolean parity function for $n\geq 3$, which is but one of an exponential number of data structures that such a QNN is unable to express. Mapping a QNN to a classical perceptron simplifies training, allowing us to systematically study the inductive biases of other, more expressive embeddings on Boolean data. Several popular embeddings primarily produce an inductive bias towards functions with low class balance, reducing their generalisation performance compared to deep neural network architectures which exhibit much richer inductive biases. We explore two alternate strategies that move beyond standard QNNs. In the first, we use a QNN to help generate a classical DNN-inspired kernel. In the second we draw an analogy to the hierarchical structure of deep neural networks and construct a layered non-linear QNN that is provably fully expressive on Boolean data, while also exhibiting a richer inductive bias than simple QNNs. Finally, we discuss characteristics of the QNN literature that may obscure how hard it is to achieve quantum advantage over deep learning algorithms on classical data.

翻译：基于参数化量子电路的量子机器学习模型，通常称为量子神经网络（QNNs），被认为是近期量子设备应用中最具前景的候选方案之一。本文通过利用从输入为 $x$ 的 QNN 到作用于 $x \otimes x$（推广至复数输入）的经典感知机的精确映射，探索了 QNN 的表达能力与归纳偏置。感知机架构的简洁性使我们能够清晰地展示当前 QNN 模型的局限性，以及它们成为实用通用学习算法所面临的诸多障碍。例如，采用幅度编码的 QNN 无法表达 $n\geq 3$ 时的布尔奇偶校验函数，而这仅是此类 QNN 无法表达的指数级数量数据结构中的一例。将 QNN 映射为经典感知机简化了训练过程，使我们能够系统研究其他更具表达能力的嵌入方式在布尔数据上的归纳偏置。若干常用嵌入主要产生对低类别平衡函数的归纳偏置，与展现出更丰富归纳偏置的深度神经网络架构相比，这降低了它们的泛化性能。我们探讨了超越标准 QNN 的两种替代策略。第一种策略中，我们使用 QNN 辅助生成受经典 DNN 启发的核函数。第二种策略中，我们类比深度神经网络的层次结构，构建了一种分层非线性 QNN，该模型在布尔数据上被证明具有完全表达能力，同时展现出比简单 QNN 更丰富的归纳偏置。最后，我们讨论了 QNN 文献中可能掩盖其在经典数据上超越深度学习算法实现量子优势难度的若干特征。