We prove that using global observables to train the matrix product state ansatz results in the vanishing of all partial derivatives, also known as barren plateaus, while using local observables avoids this. This ansatz is widely used in quantum machine learning for learning weakly entangled state approximations. Additionally, we empirically demonstrate that in many cases, the objective function is an inner product of almost sparse operators, highlighting the potential for classically simulating such a learning problem with few quantum resources. All our results are experimentally validated across various scenarios.

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.