Graph Neural Networks for Parameterized Quantum Circuits Expressibility Estimation

TL;DR

The paper tackles the costly estimation of Parameterized Quantum Circuit expressibility by introducing a Graph Neural Network that predicts $Expr = D_{KL}(P_{PQC}(F;\vec{\theta}) || P_{Haar}(F))$ directly from graph-encoded circuit representations and backend noise information. It builds a large dataset of 25,000 noiseless and 12,000 noisy PQCs (up to 8 qubits for training, with extrapolation to 10) and uses a graph-transformer architecture to achieve RMSEs around $0.05$–$0.06$ across backends, including hardware-inspired backends. The evaluation against 19 reference circuits and 64 RealAmplitude circuits shows close alignment with ground-truth expressibility and demonstrates robustness to noise, as well as extrapolation capabilities when trained on higher-qubit data. Overall, the approach offers a scalable, efficient alternative to fidelity-distribution sampling for PQC expressibility, enabling quicker design and assessment of PQCs for VQAs on both simulators and real devices.

Abstract

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.

Figures (11)

• Figure 1: Our proposed framework for predicting expressibility of Parameterized Quantum Circuits (PQCs) using Graph Neural Networks (GNNs): First, we derive a graph representation of a PQC with input, gate, and output nodes encoded with node features, edge connectivity, global circuit features, and target backend properties. Its true expressibility value $Expr$ is computed using Equation \ref{['eq:expr']}. In the model learning phase, the GNN model takes the encoded features as inputs and adjusts its parameters to minimize the difference between predicted and true expressibility values.
• Figure 2: Node feature vector (23 digits): six digits for node type, ten digits for target qubit(s), and seven digits for backend calibration Information.
• Figure 3: Overview of the GNN model for expressibility prediction of PQC. The node features are passed to the graph transformer layer to capture the neighboring correlations from nodes. Global features are propagated through fully connected layers and then concatenated with the feature vector from transformer layers. The aggregated feature vector is then directed to fully connected regressor layers to perform the regression task of expressibility prediction for PQCs.
• Figure 4: Learning curves of our proposed GNN model on training and validation dataset of randomly generated PQCs on noiseless simulators and three noisy backends (FakeGuadalupe, FakeHanoi, and FakeMumbai). Each plot demonstrates consistent learning across different quantum backends.
• Figure 5: Scatter plots of expressibility for randomly generated PQCs on noiseless simulators and three noisy backends (FakeGuadalupe, FakeHanoi, and FakeMumbai). Our GNN-based prediction model can provide close to accurate estimations of expressibility with RMSE $0.05$ in a noiseless IBM QASM simulator and RMSE lower than $0.06$ in the three noisy backends.