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

Youla Yang

Published Jan 29, 2026• 0 citations
Publisher

Abstract

Abstract 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.

DOI
10.21203/rs.3.rs-8253544/v1

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•Jan 1, 2023•0 citations
Abstract 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 Ax=bA\boldsymbol{x} = \boldsymbol{b}Ax=b as a graph with AAA encoded in the edges and b\boldsymbol{b}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∈[4,10]n\in[4,10]n∈[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.
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.
Variational Quantum Algorithms (VQAs) have gained prominence as a viable framework for exploiting near-term quantum devices in applications ranging from optimization and chemistry simulation to machine learning. However, the effectiveness of VQAs is often constrained by the so-called barren plateau problem, wherein gradients diminish exponentially as system size or circuit depth increases, thereby hindering training. In this work, we propose a reinforcement learning (RL)-based initialization strategy to alleviate the barren plateau issue by reshaping the initial parameter landscape to avoid regions prone to vanishing gradients. In particular, we explore several RL algorithms (Deterministic Policy Gradient, Soft ActorCritic, and Proximal Policy Optimization, etc.) to generate the circuit parameters (treated as “actions”) that minimize the VQAs cost function before standard gradient-based optimization. By pre-training with RL in this manner, subsequent optimization using methods such as gradient descent or Adam proceeds from a more favorable initial state. Extensive numerical experiments under various noise conditions and tasks consistently demonstrate that the RL-based initialization method significantly enhances both convergence speed and final solution quality. Moreover, comparisons among different RL algorithms highlight that multiple approaches can achieve comparable performance gains, underscoring the flexibility and robustness of our method. These findings shed light on a promising avenue for integrating machine learning techniques into quantum algorithm design, offering insights into how RL-driven parameter initialization can accelerate the scalability and practical deployment of VQAs. Opening up a promising path for the research community in machine learning for quantum, especially barren plateau problems in VQAs.
Abstract Combining classical optimization with parameterized quantum circuit evaluation, variational quantum algorithms (VQAs) are among the most promising algorithms in near-term quantum computing. However, the training of variational quantum circuits (VQCs) is susceptible to a phenomenon known as barren plateaus (BPs). Various approaches have been proposed to mitigate this issue, such as using neural networks to generate VQC parameters. In this paper, we improve the NN-based BP mitigation approach by refining the neural network architecture and extend its applicability to a more generalized scenario that includes random quantum inputs and VQC structures. We evaluate the effectiveness of the approach by comparing the convergence speed before and after it is utilized. Furthermore, we give an explanation for the effectiveness of the NN-based BP mitigation approach by utilizing a loss landscape visualization technique and the expressibility metric of VQC. The smoothness of the loss landscape offers an intuitive insight into the approach’s utility, while the reduction in expressibility accounts for the enhanced trainability. Our research highlights the universal applicability of the NN-based BP mitigation approach, underscoring its potential to drive progress in the development of VQAs across diverse domains.
Variational Quantum Eigensolvers (VQEs) are a leading class of noisy intermediate-scale quantum (NISQ) algorithms with broad applications in quantum physics and quantum chemistry. However, as system size increases, VQE optimization is increasingly hindered by the barren plateau phenomenon, where gradients vanish and the loss function becomes trapped in local minima. While machine learning-based parameter initialization methods have been proposed to address this challenge, they often show limited effectiveness in complex VQE problems. This is primarily due to their inadequate ability to model the intricate correlations embedded in the Hamiltonian structure and the associated ansatz circuits. In this paper, we propose Qracle, a graph neural network (GNN)-based parameter initializer for VQEs. Qracle systematically encodes both the Hamiltonian and the associated ansatz circuit into a unified graph representation and leverages a GNN to learn a mapping from VQE problem graphs to optimized ansatz parameters. Compared to state-of-the-art initialization techniques, Qracle achieves a reduction in initial loss of up to 10.86, accelerates convergence by decreasing optimization steps by up to 64.42%, and improves final performance with up to a 26.43% reduction in Symmetric Mean Absolute Percentage Error (SMAPE), with all results averaged over the validation set of each respective application. The source code is publicly available at https://github.com/chizhang24/Qracle.
Variational Quantum Eigensolvers (VQEs) are a leading class of noisy intermediate-scale quantum (NISQ) algorithms with broad applications in quantum physics and quantum chemistry. However, as system size increases, VQE optimization is increasingly hindered by the barren plateau phenomenon, where gradients vanish and the loss function becomes trapped in local minima. While machine learning-based parameter initialization methods have been proposed to address this challenge, they often show limited effectiveness in complex VQE problems. This is primarily due to their inadequate ability to model the intricate correlations embedded in the Hamiltonian structure and the associated ansatz circuits. In this paper, we propose \textit{Qracle}, a graph neural network (GNN)-based parameter initializer for VQEs. \textit{Qracle} systematically encodes both the Hamiltonian and the associated ansatz circuit into a unified graph representation and leverages a GNN to learn a mapping from VQE problem graphs to optimized ansatz parameters. Compared to state-of-the-art initialization techniques, \textit{Qracle} achieves a reduction in initial loss of up to 10.8610.8610.86, accelerates convergence by decreasing optimization steps by up to 64.42%64.42\%64.42%, and improves final performance with up to a 26.43%26.43\%26.43% reduction in Symmetric Mean Absolute Percentage Error (SMAPE).
Recent advancements in machine learning have revolutionised research across various fields. Despite their success, conventional learning techniques are hindered by their significant computational resources and energy requirements. Prompted by recent experimental breakthroughs in quantum computing, variational quantum machine learning (QML) – machine learning integrated with variational quantum circuits (VQCs) – has emerged as a promising alternative. Nonetheless, the theoretical framework underpinning the advantages of variational QML is still rudimentary. Specifically, the training of VQCs faces several challenges, such as the barren plateau problem, where the gradient diminishes exponentially with an increasing qubits. A related issue arises in variational QML training, where the convergence rate is exponentially small. In this thesis, we present theoretically guaranteed solutions to these challenges. First, we construct innovative circuit architectures to address the vanishing gradient problem in deep VQCs. We propose quantum controlled-layer and quantum ResNet structures, demonstrating that the expected gradient norm's lower bound is unaffected by the increase in qubits and circuit depth. Next, we introduce an initialization strategy to mitigate the vanishing gradient issue in general deep quantum circuits. We prove that Gaussian-initialized parameters ensure the gradient norm's decay rate remains inversely polynomial despite the increase in qubit numbers and circuit depth. Finally, we propose a novel and effective theory for analysing the training of quantum neural networks with moderate depths. We prove that, under certain randomness conditions in the circuits and datasets, training converges linearly with a rate inversely proportional to the dataset size. Our approach surpasses previous results, achieving exponentially larger convergence rates with modest depth, or conversely, requiring exponentially less depth for equivalent rates.