How Measurement Noise Affects Saddle Point Escape in Variational Quantum Algorithms

Stochastic gradient descent (SGD) is a widely used optimization method in both classical machine learning and the Variational Quantum Eigensolver (VQE). It plays a crucial role in overcoming saddle points and escaping local minima. In the context of VQE implemented on quantum hardware, measurement noise is unavoidable due to the finite number of measurements used to estimate expectation values. As a result, the gradient descent in VQE naturally incorporates stochasticity, leading to an SGD-like behavior. Despite its importance, the precise structure and effects of measurement noise in VQE remain unclear, particularly how it influences the dynamics of escaping saddle points and local minima. In this work, we explore the properties of measurement noise in VQE optimization, specifically its role in escaping saddle points. Our analysis shows that as measurement noise increases, the escape time decreases following a power-law. Moreover, we observe that the escape time remains nearly constant when the ratio of the learning rate η to the number of measurements Ns is kept fixed. This scaling is consistent with predictions from a stochastic differential equation (SDE), which serves as a continuous-time approximation of SGD. These findings suggest that insights into VQE optimization dynamics can be gained from studying the simpler SDE model, which closely mirrors the discrete-time SGD behavior.