  Parameterized quantum circuits (PQCs) are fundamental to quantum machine
learning (QML), quantum optimization, and variational quantum algorithms
(VQAs). The expressibility of PQCs is a measure that determines their
capability to harness the full potential of the quantum state space. It is thus
a crucial guidepost to know when selecting a particular PQC ansatz. However,
the existing technique for expressibility computation through statistical
estimation requires a large number of samples, which poses significant
challenges due to time and computational resource constraints. This paper
introduces a novel approach for expressibility estimation of PQCs using Graph
Neural Networks (GNNs). We demonstrate the predictive power of our GNN model
with a dataset consisting of 25,000 samples from the noiseless IBM QASM
Simulator and 12,000 samples from three distinct noisy quantum backends. The
model accurately estimates expressibility, with root mean square errors (RMSE)
of 0.05 and 0.06 for the noiseless and noisy backends, respectively. We compare
our model's predictions with reference circuits [Sim and others, QuTe'2019] and
IBM Qiskit's hardware-efficient ansatz sets to further evaluate our model's
performance. Our experimental evaluation in noiseless and noisy scenarios
reveals a close alignment with ground truth expressibility values, highlighting
the model's efficacy. Moreover, our model exhibits promising extrapolation
capabilities, predicting expressibility values with low RMSE for out-of-range
qubit circuits trained solely on only up to 5-qubit circuit sets. This work
thus provides a reliable means of efficiently evaluating the expressibility of
diverse PQCs on noiseless simulators and hardware.
