  Parameterized Quantum Circuits (PQCs) are essential to quantum machine
learning and optimization algorithms. The expressibility of PQCs, which
measures their ability to represent a wide range of quantum states, is a
critical factor influencing their efficacy in solving quantum problems.
However, the existing technique for computing expressibility relies on
statistically estimating it through classical simulations, which requires many
samples. In this work, we propose a novel method based on Graph Neural Networks
(GNNs) for predicting the expressibility of PQCs. By leveraging the graph-based
representation of PQCs, our GNN-based model captures intricate relationships
between circuit parameters and their resulting expressibility. We train the GNN
model on a comprehensive dataset of PQCs annotated with their expressibility
values. Experimental evaluation on a four thousand random PQC dataset and IBM
Qiskit's hardware efficient ansatz sets demonstrates the superior performance
of our approach, achieving a root mean square error (RMSE) of 0.03 and 0.06,
respectively.
