Numerical Optimization Strategies for the Variational Hamiltonian Ansatz in Noisy Quantum Environments

TL;DR

This paper tackles the challenge of optimizing variational quantum eigensolvers in noisy environments by benchmarking eight classical optimizers on the truncated Variational Hamiltonian Ansatz (tvha) across H$_2$, H$_4$, and LiH. By comparing statevector and finite-shot simulations, it shows that gradient-based methods excel in noiseless settings while population-based approaches, especially CMA-ES, offer superior robustness under sampling noise; Hartree–Fock initialization helps small systems but its advantage wanes with system size. The authors introduce and leverage the concept of a sampling-noise floor, demonstrating that evolution strategies converge to a steady-state distribution defined by this floor, and that averaging over populations can yield energy estimates beyond the intrinsic sampling limit. They provide practical guidelines for optimizer choice, initialization, and active-space truncation to improve reliability on near-term quantum devices. The work highlights the central role of sampling noise in determining optimization performance and outlines a path toward more accurate energy estimates via population-based averaging and careful resource allocation.

Abstract

The prevalence of variational methods in near-term quantum computing makes optimizer choice critical, yet selection is frequently intuition-based. We therefore present a systematic benchmark of eight classical optimization algorithms for variational quantum chemistry using the truncated Variational Hamiltonian Ansatz. Performance is evaluated on H$_2$, H$_4$, and LiH in both full and active-space representations under noiseless and finite-shot sampling noise. Sampling noise substantially reshapes cost landscapes, induces wandering near minima, and flips optimizer rankings: gradient-based methods perform best in noiseless simulations, whereas population-based optimizers, particularly CMA-ES, show greater robustness under finite-shot noise. Optimizer performance is strongly problem dependent: Hartree-Fock initialization aids small systems, but its advantage diminishes with system size. Also, we observe that finite shot sampling frequently violates the lower bound given by the variational principle, a principle that cannot be strictly held in the presence of noise. By exploiting the guaranteed convergence of Evolution Strategies to a steady state distribution defined by the noise floor, we utilize the symmetry of these violations to achieve energy estimation precision beyond the intrinsic sampling limit.

Figures (19)

• Figure 1: The figure illustrates a fully decomposed 4-qubit tvha with 3 trainable parameters, generated for finding the ground state of the H2 molecule. The circuit is constructed using a gate set of {CX (48), H (32), $\sqrt{\textbf{X}}$ (16), RZ (16), $\sqrt{\textbf{X}}^{\dagger}$ (16), RZZ (6), U3 (2)}. This gate configuration effectively balances chemical accuracy with nisq-era hardware limitations.
• Figure 2: Energy landscapes for the H2 molecule under varying levels of sampling noise. (a) shows the smooth contours characteristic of exact statevector simulation, revealing quasi-degenerate valleys. (b) reveals emerging distortions, particularly in contour line warping, with moderate noise. (c) highlights key phenomena under significant noise: false minima (blue/purple regions below $E_0$), gradient reversals, and noise induced ruggedness within quasidegenerate valleys, where the sampling noise creates artificial local minima specifically along the flat valley floor.
• Figure 3: Energy error progression for H2 using tvha with cmaes population size 25. Top: fe of all individuals (colored points), average of fe in iteration (black crosses), best (lowest) fe in iteration (red crosses) and noise floor (red dashed lines). Bottom: Aggregated absolute errors for the mean of fe in iteration (black line), aggregated absolute errors for the best fe in iteration (red line), and the best iteration average (blue) approaches compared to the computed noise floor error (purple).
• Figure 4: Convergence of optimization algorithms for H2 energy on statevector and sampling simulations. The plot shows the mean energy over 10 independent runs as a function of function evaluations (log-log scale). Crosses mark the lowest energies encountered during optimization, while circles denote high-shot ($10^{5}$) reevaluations of the corresponding parameters.
• Figure 5: Convergence of optimization algorithms for H4 energy on statevector and sampling simulations. The plot shows the mean energy over 10 independent runs as a function of function evaluations (log-log scale). Crosses mark the lowest energies encountered during optimization, while circles denote high-shot ($10^{5}$) reevaluations of the corresponding parameters.