PVLS: A Learning-based Parameter Prediction Technique for Variational Quantum Linear Solvers

TL;DR

We address barren plateaus in variational quantum linear solvers (VQLS) by introducing PVLS, a graph neural network–based initializer that learns to map the structure of a linear system $A\boldsymbol{x}=\boldsymbol{b}$ to high-quality VQC parameters. PVLS encodes $(A,\boldsymbol{b})$ as a signed directed graph and uses Lap-GCN with directed aggregation to predict a $(q_n\times 3)$ parameter vector, improving convergence and reducing optimization burden. On a large dataset of >15,000 synthetic instances and ten real sparse matrices, PVLS achieves up to a 2.6× speedup in training with millisecond-scale inference per instance and substantial reductions in initial loss, demonstrating robust generalization to unseen problems. These results indicate that ML-guided initialization is a practical strategy to enhance the viability of hybrid quantum–classical workflows on NISQ devices, with potential validation on real hardware forthcoming.

Abstract

Variational Quantum Linear Solvers (VQLS) are a promising method for solving linear systems on near-term quantum devices. However, their performance is often limited by barren plateaus and inefficient parameter initialization, which significantly hinder trainability as the system size increases. In this work, we introduce PVLS, a learning-based parameter prediction framework that uses Graph Neural Networks (GNNs) to generate high-quality initial parameters for VQLS circuits. By leveraging structural information from the coefficient matrix, PVLS predicts expressive and scalable initializations that improve convergence and reduce optimization difficulty. Extensive experiments on matrix sizes ranging from 16 to 1024 show that PVLS provides up to a 2.6x speedup in optimization and requires fewer iterations while maintaining comparable solution accuracy. These results demonstrate the potential of machine-learning-guided initialization strategies for improving the practicality of hybrid quantum-classical algorithms in the NISQ era.

Figures (6)

• Figure 1: Overview of the VQLS algorithm. The input to VQLS consists of a matrix $A$ expressed as a linear combination of unitaries $A_l$ and a variational quantum circuit (VQC) $V$ that prepares the quantum state $\ket{\boldsymbol{b}}$. The output of VQLS is a quantum state $\ket{\boldsymbol{x}}$ that is approximately proportional to the solution of the linear system $A\boldsymbol{x} = \boldsymbol{b}$. The parameters $\boldsymbol{\alpha}$ in $V(\boldsymbol{\alpha})$ are iteratively optimized within a hybrid quantum--classical loop to minimize the cost function $C(\boldsymbol{\alpha})$.
• Figure 2: Graph representation of a linear system in PVLS. In Figure \ref{['f:vls_graph_encoding2']}, the sign of $a_{ij}$ determines the corresponding element in the adjacency matrix, while $|a_{ij}|$ is the weight of the edge $\{i,j\}$.
• Figure 3: Graph neural network (GNN) architecture of PVLS.
• Figure 4: Initial loss comparison between random initialization and PVLS across different matrix sizes. Each subplot shows the initial loss distribution for a fixed system size $2^n \times 2^n$ with $n \in [4,10]$.
• Figure 5: Training loss convergence of VQLS with random initialization and PVLS across different matrix sizes. Each subplot reports the mean loss and variance band over optimization iterations for a fixed system size $2^n \times 2^n$, $n \in [4,10]$.