  Variational quantum algorithms (VQAs) are the quantum analog of classical
neural networks (NNs). A VQA consists of a parameterized quantum circuit (PQC)
which is composed of multiple layers of ansatzes (simpler PQCs, which are an
analogy of NN layers) that differ only in selections of parameters. Previous
work has identified the alternating layered ansatz as potentially a new
standard ansatz in near-term quantum computing. Indeed, shallow alternating
layered VQAs are easy to implement and have been shown to be both trainable and
expressive. In this work, we introduce a training algorithm with an exponential
reduction in training cost of such VQAs. Moreover, our algorithm uses classical
shadows of quantum input data, and can hence be run on a classical computer
with rigorous performance guarantees. We demonstrate 2--3 orders of magnitude
improvement in the training cost using our algorithm for the example problems
of finding state preparation circuits and the quantum autoencoder.
