Reliable Optimization Under Noise in Quantum Variational Algorithms

TL;DR

The paper addresses how finite-shot sampling noise distorts the variational quantum eigensolver (VQE) optimization landscape, causing stochastic variational bound violations and the winner's curse. By benchmarking eight classical optimizers on problem classes including tvha-based molecular systems and TwoLocal-based Ising/Hubbard models, it develops a statistical framework with $\bar{C}(\boldsymbol{\theta}) = C(\boldsymbol{\theta}) + \epsilon_{sampling}$ and a noise floor $\sigma_{noise}$, showing gradient-based methods lose stability as noise approaches curvature scales. The study finds adaptive population-based optimizers, notably CMA-ES and iL-SHADE, to be the most robust under noise, and demonstrates that tracking population means or applying high-shot reevaluation mitigates false minima (winner's curse). Practically, the authors propose guidelines including co-design of compact ansatzes (e.g., tvha) and adaptive optimizers, along with dimensionality reduction via active space truncation, to achieve sub-millihartree accuracy in noisy VQE scenarios.

Abstract

The optimization of Variational Quantum Eigensolver is severely challenged by finite-shot sampling noise, which distorts the cost landscape, creates false variational minima, and induces statistical bias called winner's curse. We investigate this phenomenon by benchmarking eight classical optimizers spanning gradient-based, gradient-free, and metaheuristic methods on quantum chemistry Hamiltonians H$_2$, H$_4$ chain, LiH (in both full and active spaces) using the truncated Variational Hamiltonian Ansatz. We analyze difficulties of gradient-based methods (e.g., SLSQP, BFGS) in noisy regimes, where they diverge or stagnate. We show that the bias of estimator can be corrected by tracking the \textit{population mean}, rather than the biased best individual when using population based optimizer. Our findings, which are shown to generalize to hardware-efficient circuits and condensed matter models, identify adaptive metaheuristics (specifically CMA-ES and iL-SHADE) as the most effective and resilient strategies. We conclude by presenting a set of practical guidelines for reliable VQE optimization under noise, centering on the co-design of physically motivated ansatz and the use of adaptive optimizers.

Figures (5)

• Figure 1: Combined comparison of energy landscapes across models and noise levels. Columns correspond to the Ising model (left), Hubbard model (center), and H2 system (right). Rows correspond to statevector/noiseless (top), 64–512-shots (second), 5120–$6{\times}1024$-shots (third), and alternative H2 parameter slices (bottom).
• Figure 2: Energy landscape slice for noisy VQE estimation of H2 using 512 shots with tvha ansatz. The red plane indicates the exact ground-state energy $E_0$; apparent dips below $E_0$ arise from sampling noise, violating the variational principle (false minima).
• Figure 3: Energy error from all individuals for H2 using tvha. Top: Optimization trajectories (colored points), iteration means (black crosses), best values (red crosses), and noise floors (red dashed lines). Bottom: Average absolute errors for mean-based (black), best-value (red), $\sigma_{\text{noise}}$ (purple).
• Figure 4: Convergence comparison for the six-site Hubbard model using 64-shot estimation. Adaptive metaheuristics (iL-SHADE and cmaes) maintain stable progress where gradient-based and basic Differential Evolution variants stagnate.
• Figure 5: Convergence curves for H2, H4, and LiH in full and active space with random initialization starts using tvha. Finite-shot sampling increases variance and erases the performance gap between deterministic and evolutionary optimizers. In the right panel we can see optimizers convergence to dip below ground state energies (red dashed line).