Quantum machine learning requires powerful, flexible and efficiently trainable models to be successful in solving challenging problems. In this work, we present density quantum neural networks, a learning model incorporating randomisation over a set of trainable unitaries. These models generalise quantum neural networks using parameterised quantum circuits, and allow a trade-off between expressibility and efficient trainability, particularly on quantum hardware. We demonstrate the flexibility of the formalism by applying it to two recently proposed model families. The first are commuting-block quantum neural networks (QNNs) which are efficiently trainable but may be limited in expressibility. The second are orthogonal (Hamming-weight preserving) quantum neural networks which provide well-defined and interpretable transformations on data but are challenging to train at scale on quantum devices. Density commuting QNNs improve capacity with minimal gradient complexity overhead, and density orthogonal neural networks admit a quadratic-to-constant gradient query advantage with minimal to no performance loss. We conduct numerical experiments on synthetic translationally invariant data and MNIST image data with hyperparameter optimisation to support our findings. Finally, we discuss the connection to post-variational quantum neural networks, measurement-based quantum machine learning and the dropout mechanism.

翻译：量子机器学习需要强大、灵活且可高效训练的模型，才能在解决具有挑战性的问题中取得成功。在本工作中，我们提出了密度量子神经网络，这是一种包含一组可训练酉矩阵随机化的学习模型。这些模型推广了使用参数化量子电路的量子神经网络，并允许在表达能力和高效可训练性之间进行权衡，尤其是在量子硬件上。我们通过将该形式体系应用于两个最近提出的模型族，展示了其灵活性。第一种是交换块量子神经网络，其可高效训练但表达能力可能受限。第二种是正交（保持汉明权重）量子神经网络，它们能对数据提供定义明确且可解释的变换，但在量子设备上难以进行大规模训练。密度交换量子神经网络以最小的梯度复杂度开销提升了容量，而密度正交神经网络则实现了从二次到常数的梯度查询优势，且性能损失极小甚至为零。我们在合成平移不变数据和MNIST图像数据上进行了数值实验，并进行了超参数优化以支持我们的发现。最后，我们讨论了与后变分量子神经网络、基于测量的量子机器学习以及丢弃机制之间的联系。