Expressive power of parametrized quantum circuits

Abstract

Parameterized quantum circuits (PQCs) have been broadly used as a hybrid\nquantum-classical machine learning scheme to accomplish generative tasks.\nHowever, whether PQCs have better expressive power than classical generative\nneural networks, such as restricted or deep Boltzmann machines, remains an open\nissue. In this paper, we prove that PQCs with a simple structure already\noutperform any classical neural network for generative tasks, unless the\npolynomial hierarchy collapses. Our proof builds on known results from tensor\nnetworks and quantum circuits (in particular, instantaneous quantum polynomial\ncircuits). In addition, PQCs equipped with ancillary qubits for post-selection\nhave even stronger expressive power than those without post-selection. We\nemploy them as an application for Bayesian learning, since it is possible to\nlearn prior probabilities rather than assuming they are known. We expect that\nit will find many more applications in semi-supervised learning where prior\ndistributions are normally assumed to be unknown. Lastly, we conduct several\nnumerical experiments using the Rigetti Forest platform to demonstrate the\nperformance of the proposed Bayesian quantum circuit.