Classical Regularization in Variational Quantum Eigensolvers

TL;DR

The paper tackles instability and poor conditioning in Variational Quantum Eigensolver optimization due to barren plateaus. It proposes a purely classical remedy: add a quadratic $L_2^2$ penalty to the objective, $ ilde{E}(oldsymbol{ heta}) = E(oldsymbol{ heta}) + oldsymbol{\lambda} \big\\|oldsymbol{\theta}\boldsymbol\_2\big\ig^2$, without modifying the quantum circuit, and validates it with a two-stage cosine-decay schedule. Across H$_2$, LiH, and RFIM, a broad moderate regularization window improves convergence reliability, reduces parameter-norm variance, and preserves ground-state energies. A practical $oldsymbol{\lambda_{ ext{scale}}}$ heuristic guides initialization, and the approach remains robust across optimizers, indicating broad applicability to hybrid quantum–classical optimization without circuit changes.

Abstract

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 proportional to the squared norm of the parameters, without modifying the quantum circuit or measurement process. Across all tested Hamiltonians, H2, LiH, and the Random Field Ising Model (RFIM), we observe improved performance over a broad window of the regularization strength. 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.

Figures (4)

• Figure 1: Example of the 4-qubit hardware-efficient TwoLocal ansatz used for H$_2$ (Ry/Rz blocks). The experiment used an initial rotation layer followed by 4 layers, meanwhile in this figure only 2 layers are depicted for optimal visual clarity.
• Figure 2: H$_2$ molecule: success rate versus $\lambda$ for different chemical-accuracy thresholds. Panels (a)–(f) show the fraction of runs reaching $\Delta E \le 1.5\times10^{-k}$ Ha for decreasing values of $k$. A broad moderate-regularization window yields consistently high success rates across thresholds.
• Figure 3: LiH molecule: success rate versus $\lambda$ for two chemical-accuracy thresholds. The stabilizing window shifts to smaller $\lambda$ values compared with H$_2$, reflecting increased dimensionality and stronger redundant directions in the ansatz.
• Figure 4: RFIM benchmark: success rate versus $\lambda$ for two chemical-accuracy thresholds. The stabilizing window lies in the broad interval $\lambda \approx 0.05$--$0.12$, and moderate regularization consistently improves convergence reliability across this strongly nonconvex landscape.