Classical Regularization Achieves Stable Convergence for Variational Quantum Eigensolvers

Scientists are tackling the persistent challenge of instability in variational quantum algorithms, a key hurdle in harnessing the power of near-term quantum computers. Yury Chernyak, Ijaz Ahamed Mohammad, and Martin Plesch, all from the Institute of Physics, Slovak Academy of Sciences and Matej Bel University, demonstrate a surprisingly effective solution: classical regularization. Their research reveals that simply adding a standard L2 squared-norm penalty to the optimisation process significantly stabilises hybrid quantum-classical optimisation across diverse systems , including H2, LiH, and the Random Field Ising Model. This work is significant because it offers a robust, system-independent method to improve the reliability and reproducibility of variational quantum optimisation without requiring changes to the quantum hardware or circuits themselves, paving the way for more dependable results from today’s quantum devices.

L2 Regularization Stabilises Variational Quantum Algorithms by preventing

Scientists have demonstrated a robust method for stabilizing variational quantum algorithms (VQAs) using a purely classical approach, L2 squared-norm regularization. The research establishes a universal, non-monotonic dependence on the regularization strength, identifying a broad window where convergence significantly improves, parameter norms shrink, and the probability of successful optimization rises sharply. By incorporating a penalty term, R(θ) = λ∥θ∥2, into the VQE objective, they biased the optimizer towards smoother regions of parameter space, effectively reshaping the optimization landscape. This work opens new avenues for enhancing the performance of near-term quantum computers, which are currently limited in both size and quality.
The study unveils that by strategically conditioning the Classical Optimization landscape, it is possible to overcome some of the fundamental obstacles hindering VQAs, such as vanishing gradients and sensitivity to initial conditions. The team’s large-scale numerical results demonstrate that this classical remedy is not only effective but also remarkably robust, functioning consistently across different quantum systems. This discovery is particularly significant as it allows for improvements to VQE without requiring modifications to the quantum hardware or the quantum circuit design itself. Furthermore, the geometric motivation behind this approach suggests that L2 regularization improves conditioning by suppressing parameter directions that contribute little to the underlying model but introduce significant curvature into the objective function. This reshaping of the optimization landscape biases the search towards more stable and reliable solutions, ultimately enhancing the efficiency and accuracy of VQE.

L2 Regularization for VQE Optimisation Stability improves performance

Scientists investigated a classical remedy, L2 squared-norm regularization, to stabilize hybrid quantum-classical optimization within Variational Quantum Algorithms (VQAs). To disentangle the effects of regularization from final convergence, experiments employed a two-stage optimization structure. Stage A utilized a regularized exploration phase, promoting stable search, while Stage B performed an unregularized refinement to polish the solution, the total function-evaluation budget was shared between these stages. The larger LiH system necessitated the limited-memory BFGS optimizer (L-BFGS-B) with an increased iteration cap of 200 per stage to accommodate its 80 parameters.
The study pioneered a two-stage optimization scheme with a cosine-decay scheduler to isolate and quantify the stabilizing role of classical regularization. In Stage A, the regularization term decayed smoothly following the equation λ(t) = λ0 / (2 + cos(πt/TA)), where λ0 represents the initial regularization strength and TA denotes the number of iterations in Stage A. This ensured a gradual reduction of the penalty as the optimizer approached a stable region, transitioning seamlessly into the unregularized refinement phase of Stage B, which resumed from the best parameters of Stage A with λ = 0. This design enabled precise measurement of regularization’s effect during exploration while maintaining unbiased final energy estimates.

Researchers systematically scanned initial regularization strengths (λ0) across the interval {0.00, 0.005, 0.020, 0.050, 0.075, 0.099, 0.125, 0.150, 0.175, 0.200, 0.250} to identify the optimal range. They explored each setting over thousands of random initializations, 10,000 runs for H2, 6,000 for LiH, and 10,000 for RFIM, recording success rates, median final energies, and parameter-norm statistics. The team defined λopt as the contiguous range of λ values achieving a mean success rate of at least 90% of the global maximum at a defined target accuracy, establishing a stable performance plateau. Experiments were executed with 15 parallel processes using Python 3.13, NumPy 2.3, SciPy 1.16, and Qiskit 2.1, leveraging the noiseless StatevectorEstimator to isolate algorithmic effects and utilizing the Devana supercomputer for the complex LiH system.

L2 Regularization Stabilises Variational Quantum Eigensolver optimisation

Scientists achieved a significant breakthrough in stabilizing hybrid quantum-classical algorithms through the implementation of standard L2 squared-norm regularization. Experiments revealed a universal, non-monotonic dependence on the regularization coefficient λ, with a substantial stabilization window identified where convergence improved markedly. Data shows that within this window, parameter norms shrank, and success probabilities rose sharply, indicating a significant enhancement in optimization performance. Specifically, the study incorporated a penalty term R(θ) = λ∥θ∥2 into the classical objective, leaving the quantum circuit unchanged while biasing the optimizer towards smoother regions of parameter space.

Results demonstrate that the L2 regularization reshapes the geometry of the objective function, biasing the search towards better-conditioned regions without altering the physical location of the ground-state minimum. The team’s work highlights the geometric effect of regularization, suppressing oscillatory modes that complicate gradient-based search and improving the conditioning of the classical optimization landscape. Further research will likely explore the application of this technique to other variational quantum algorithms and more complex quantum systems, potentially unlocking new possibilities in quantum simulation and optimization.

L2 Regularization Stabilises Variational Quantum Optimisation

Scientists have demonstrated that classical L2 squared-norm regularization can systematically stabilize optimization in variational quantum algorithms (VQAs). The research reveals a non-monotonic relationship between regularization strength and optimization performance, establishing a ‘stabilization window’ where benefits are maximized. Smaller regularization values maintain optimization freedom, while intermediate values condition the variational landscape for more robust descent toward the minimum energy. Larger values, however, can over-constrain the optimization, limiting achievable accuracy. Importantly, the optimal regularization strength is linked to the desired accuracy threshold, suggesting a trade-off between exploration and landscape conditioning.

The authors acknowledge that the optimal regularization strength is not universal and depends on the target accuracy. Furthermore, while conjugate-gradient and L-BFGS optimizers were tested, the observed benefits of classical regularization appear to be generic features of the variational landscape, rather than specific to the chosen optimizer. Future research could explore adaptive regularization schemes that dynamically adjust the regularization strength during optimization, potentially further improving performance and robustness, though this remains an area for continued investigation.