Variational hybrid quantum classical algorithms are a class of quantum algorithms intended to run on noisy intermediate-scale quantum (NISQ) devices. These algorithms employ a parameterized quantum circuit (ansatz) and a quantum-classical feedback loop. A classical device is used to optimize the parameters in order to minimize a cost function that can be computed far more efficiently on a quantum device. The cost function is constructed such that finding the ansatz parameters that minimize its value, solves some problem of interest. We focus specifically on the variational quantum linear solver, and examine the effect of several gradient-free and gradient-based classical optimizers on performance. We focus on both the average rate of convergence of the classical optimizers studied, as well as the distribution of their average termination cost values, and how these are affected by noise. Our work demonstrates that realistic noise levels on NISQ devices present a challenge to the optimization process. All classical optimizers appear to be very negatively affected by the presence of realistic noise. If noise levels are significantly improved, there may be a good reason for preferring gradient-based methods in the future, which performed better than the gradient-free methods with only shot-noise present. The gradient-free optimizers, simultaneous perturbation stochastic approximation (SPSA) and Powell’s method, and the gradient-based optimizers, AMSGrad and BFGS performed the best in the noisy simulation, and appear to be less affected by noise than the rest of the methods. SPSA appears to be the best performing method. COBYLA, Nelder–Mead and Conjugate-Gradient methods appear to be the most heavily affected by noise, with even slight noise levels significantly impacting their performance.

变分混合量子经典算法是一类旨在在嘈杂的中尺度量子 (NISQ) 设备上运行的量子算法。这些算法采用参数化量子电路（ansatz）和量子经典反馈回路。经典设备用于优化参数，以最小化可以在量子设备上更有效地计算的成本函数。构建成本函数，以便找到最小化其值的 ansatz 参数，解决一些感兴趣的问题。我们特别关注变分量子线性求解器，并检查几个无梯度和基于梯度的经典优化器对性能的影响。我们专注于研究的经典优化器的平均收敛速度，以及它们的平均终止成本值的分布，以及它们如何受噪声影响。我们的工作表明，NISQ 设备上的实际噪声水平对优化过程提出了挑战。所有经典优化器似乎都受到现实噪声的负面影响。如果噪声水平显着改善，未来可能有充分的理由更喜欢基于梯度的方法，这种方法比仅存在散粒噪声的无梯度方法表现更好。无梯度优化器、同步扰动随机逼近 (SPSA) 和 Powell 方法以及基于梯度的优化器 AMSGrad 和 BFGS 在噪声模拟中表现最好，并且似乎比其他方法受噪声影响更小。SPSA 似乎是性能最好的方法。科比拉，