Classical Regularization in Variational Quantum Eigensolvers

While quantum computers are a very promising tool for the far future, in their current state of the art they remain limited both in size and quality. This has given rise to hybrid quantum--classical algorithms, where the quantum device performs only a small but vital part of the overall computation. Among these, variational quantum algorithms (VQAs), which combine a classical optimization procedure with quantum evaluation of a cost function, have emerged as particularly promising. However, barren plateaus and ill-conditioned optimization landscapes remain among the primary obstacles faced by VQAs, often leading to unstable convergence and high sensitivity to initialization. Motivated by this challenge, we investigate whether a purely classical remedy---standard L2 squared-norm regularization---can systematically stabilize hybrid quantum--classical optimization. Specifically, we augment the Variational Quantum Eigensolver (VQE) objective with a quadratic penalty \((R(\boldsymbol{\theta}) = \lambda\|\boldsymbol{\theta}\|^2)\), without modifying the quantum circuit or measurement process. Across all tested Hamiltonians---H\((_2)\), LiH, and the Random Field Ising Model (RFIM)---we observe improved performance over a broad window of the regularization strength \((\lambda)\). Our large-scale numerical results demonstrate that classical regularization provides a robust, system-independent mechanism for mitigating VQE instability, enhancing the reliability and reproducibility of variational quantum optimization without altering the underlying quantum circuit.