We investigate the potential of near-term quantum algorithms for solving partial differential equations (PDEs), focusing on a linear one-dimensional advection-diffusion equation as a test case. This study benchmarks a ground-state algorithm, Variational Quantum Eigensolver (VQE), against three leading quantum dynamics algorithms, Trotterization, Variational Quantum Imaginary Time Evolution (VarQTE), and Adaptive Variational Quantum Dynamics Simulation (AVQDS), applied to the same PDE on small quantum hardware. While Trotterization is fully quantum, VarQTE and AVQDS are variational algorithms that reduce circuit depth for noisy intermediate-scale quantum (NISQ) devices. However, hardware results from these dynamics methods show sizable errors due to noise and limited shot statistics. To establish a noise-free performance baseline, we implement the VQE-based solver on a noiseless statevector simulator. Our results show VQE can reach final-time infidelities as low as ${O}(10^{-9})$ with $N=4$ qubits and moderate circuit depths, outperforming hardware-deployed dynamics methods that show infidelities $\gtrsim 10^{-2}$. By comparing noiseless VQE to shot-based and hardware-run algorithms, we assess their accuracy and resource demands, providing a baseline for future quantum PDE solvers. We conclude with a discussion of limitations and potential extensions to higher-dimensional, nonlinear PDEs relevant to engineering and finance.

翻译：本研究探讨近期量子算法在求解偏微分方程（PDE）方面的潜力，以一维线性平流-扩散方程作为测试案例。研究将基于基态求解的变分量子本征求解器（VQE）与三种主流量子动力学算法——Trotter分解、变分量子虚时间演化（VarQTE）及自适应变分量子动力学模拟（AVQDS）进行基准比较，这些算法均在小型量子硬件上应用于同一偏微分方程。Trotter分解为全量子算法，而VarQTE与AVQDS作为变分算法，可降低噪声中等规模量子（NISQ）设备的电路深度。然而，受噪声与有限采样统计影响，这些动力学方法在硬件上的运行结果存在显著误差。为建立无噪声性能基准，我们在无噪声态矢量模拟器上实现了基于VQE的求解器。结果表明，在$N=4$量子比特及中等电路深度条件下，VQE能达到低至${O}(10^{-9})$的终时保真度误差，优于硬件部署动力学方法所显示的$\gtrsim 10^{-2}$误差水平。通过对比无噪声VQE与基于采样及硬件运行的算法，我们评估了其精度与资源需求，为未来量子偏微分方程求解器提供了基准参考。最后，我们讨论了当前方法的局限性及其向工程与金融领域相关的高维非线性偏微分方程拓展的潜力。