PVLS: A Learning-Based Parameter Initialization Method for Variational Quantum Linear Solvers

Variational quantum linear solvers (VQLSs) are a promising class of hybrid quantum--classical algorithms for solving linear systems on near-term quantum devices. However, the performance of VQLSs is often impeded by barren plateaus, particularly when using randomly initialized variational quantum circuits (VQCs). To mitigate this issue, we propose \textit{PVLS}, a GNN-based parameter initialization framework that improves both convergence speed and final solution quality. By reformulating the linear system $A\boldsymbol{x} = \boldsymbol{b}$ as a graph with $A$ encoded in the edges and $\boldsymbol{b}$ as node features, PVLS learns to predict effective initial VQC parameters. Our method is trained on thousands of randomly generated matrices with varying dimensions ($n\in[4,10]$), using optimized VQC parameters as ground-truth labels. On unseen test instances, PVLS reduces the initial cost by an average of 81.3\% and the final loss by 71\% compared to random initialization. PVLS also accelerates convergence, reducing the number of optimization steps by more than 60\% on average. We further evaluate PVLS on ten real-world sparse matrices, demonstrating its generalization capability and robustness. Our results highlight the utility of machine-learned priors in improving the trainability of VQLSs and alleviating optimization challenges in variational quantum algorithms.