物理上可实现的量子神经网络
Toward Physically Realizable Quantum Neural Networks
摘要：
最近，人们对量子神经网络（QNN）及其在不同领域的应用产生了浓厚的兴趣。QNN的当前解决方案对它们的可扩展性提出了重大挑战，确保满足量子力学的假设，并且网络是物理可实现的。QNN的指数状态空间对训练过程的可扩展性提出了挑战。不可克隆原则禁止对训练样本进行多个拷贝，而测量假设导致了非确定性损失函数。因此，依赖于重复测量每个样本的几个副本来训练QNN的现有方法的物理可实现性和效率尚不清楚。本文提出了一种新的QNN模型，该模型依赖于量子感知器（QP）传递函数的带限傅立叶展开来设计可扩展的训练过程。该训练过程通过随机量子随机梯度下降技术进行增强，从而消除了对样本复制的需要。我们表明，这种训练过程收敛到真正的最小值的期望，即使在存在的非确定性，由于量子测量。我们的解决方案有许多重要的好处：（i）使用具有集中傅立叶功率谱的QP，我们表明QNN的训练过程可以扩展;（ii）它消除了对重新排序的需要，从而与无克隆规则保持一致;（iii）提高了整个训练过程的数据效率，因为每个数据样本每个时期处理一次。我们提出了一个详细的理论基础，我们的模型和方法的可扩展性，准确性和数据效率。我们还通过一系列的数值实验验证了我们的方法的实用性。
英文摘要：
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.
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