There has been significant recent interest in quantum neural networks (QNNs),
along with their applications in diverse domains. Current solutions for QNNs
pose significant challenges concerning their scalability, ensuring that the
postulates of quantum mechanics are satisfied and that the networks are
physically realizable. The exponential state space of QNNs poses challenges for
the scalability of training procedures. The no-cloning principle prohibits
making multiple copies of training samples, and the measurement postulates lead
to non-deterministic loss functions. Consequently, the physical realizability
and efficiency of existing approaches that rely on repeated measurement of
several copies of each sample for training QNNs are unclear. This paper
presents a new model for QNNs that relies on band-limited Fourier expansions of
transfer functions of quantum perceptrons (QPs) to design scalable training
procedures. This training procedure is augmented with a randomized quantum
stochastic gradient descent technique that eliminates the need for sample
replication. We show that this training procedure converges to the true minima
in expectation, even in the presence of non-determinism due to quantum
measurement. Our solution has a number of important benefits: (i) using QPs
with concentrated Fourier power spectrum, we show that the training procedure
for QNNs can be made scalable; (ii) it eliminates the need for resampling, thus
staying consistent with the no-cloning rule; and (iii) enhanced data efficiency
for the overall training process since each data sample is processed once per
epoch. We present a detailed theoretical foundation for our models and methods'
scalability, accuracy, and data efficiency. We also validate the utility of our
approach through a series of numerical experiments.

本文提出了一种基于带限傅里叶展开的量子感知器（QPs）转移函数的模型，用于设计可扩展的训练过程的量子神经网络（QNNs），并在其中添加了一种随机化的量子随机梯度下降技术，消除了样本复制的需要，并证明该训练过程期望收敛于真实最小值，这有助于提高数据效率和遵守禁止复制规则。