Saddle points constitute a crucial challenge for first-order gradient descent algorithms. In notions of classical machine learning, they are avoided for example by means of stochastic gradient descent methods. In this work, we provide evidence that the saddle points problem can be naturally avoided in variational quantum algorithms by exploiting the presence of stochasticity. We prove convergence guarantees and present practical examples in numerical simulations and on quantum hardware. We argue that the natural stochasticity of variational algorithms can be beneficial for avoiding strict saddle points, i.e., those saddle points with at least one negative Hessian eigenvalue. This insight that some levels of shot noise could help is expected to add a new perspective to notions of near-term variational quantum algorithms.

翻译：鞍点构成一阶梯度下降算法的主要挑战。在经典机器学习概念中，可通过随机梯度下降等方法规避鞍点问题。本文证明，通过利用随机性特性，变分量子算法可自然避免鞍点问题。我们提供了收敛性保证，并在数值模拟与量子硬件上展示了实际案例。研究表明，变分算法的固有随机性有助于规避严格鞍点（即至少存在一个负海森矩阵特征值的鞍点）。这一关于适当水平的散粒噪声可能发挥积极作用的洞见，将为近期的变分量子算法研究提供新的视角。