The Expressive Power of Parameterized Quantum Circuits

TL;DR

This paper demonstrates that simple parameterized quantum circuits outperform classical neural networks in generative tasks, especially when augmented with ancillary qubits for post-selection, with implications for Bayesian and semi-supervised learning.

Contribution

It proves the superior expressive power of certain PQCs over classical neural networks and explores their application in Bayesian learning with numerical validation.

Findings

PQCs outperform classical neural networks in generative tasks unless the polynomial hierarchy collapses.

Ancillary qubits for post-selection enhance the expressive power of PQCs.

Numerical experiments validate the effectiveness of Bayesian quantum circuits on Rigetti platform.

Abstract

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