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.

量子机器学习需要强大、灵活和高效可训练的模型来成功解决具有挑战性的问题。本文介绍了密度量子神经网络，一种融合了一组可训练酉矩阵的随机化学习模型。这些模型使用参数化量子电路广义化了量子神经网络，并允许在表达能力和高效可训练性之间进行折中，特别适用于量子硬件。我们通过将其应用于两个最近提出的模型族来展示该形式化方法的灵活性。第一个是具有有效可训练性但表达能力可能受限的交换块量子神经网络 (QNNs)。第二个是正交 (保持汉明权重) 量子神经网络，它在数据上提供了定义明确且可解释的转换，但在量子设备上进行大规模训练具有挑战性。密度交换块 QNNs 增加了容量，几乎没有梯度复杂度增加，而密度正交神经网络减少了梯度查询的复杂度，几乎没有性能损失。我们通过对具有超参数优化的合成平移不变数据和 MNIST 图像数据进行数值实验来支持我们的发现。最后，我们讨论了与后变分量量子神经网络、基于测量的量子机器学习和辍学机制之间的联系。