Abstract
The variational quantum eigensolver (VQE) is a promising algorithm for near-term quantum chemistry applications, but selecting optimal ansatz circuits remains challenging. Expressibility, a metric quantifying a circuit's ability to explore the Hilbert space, has been proposed as a guide for ansatz selection, but recent work showed it inconsistently predicts VQE performance under realistic noise for H₂. We extend this investigation to cover both H₂ and H₃⁺ under four execution scenarios: ideal, noisy, and noisy with zero-noise extrapolation (ZNE) or probabilistic error cancellation (PEC). We find that error mitigation does not reliably restore expressibility's predictive power. ZNE reduces error for only 4 of 12 H₂ circuits and 4 of 6 H₃⁺ circuits, while PEC actually increases error in 11 of 12 H₂ circuits and all 6 H₃⁺ circuits. We reproduce and extend Saib et al.'s key finding that circuit rankings scramble under noise (Spearman ρ ≈ -0.1 between ideal and noisy rankings), and identify a new result: ZNE largely preserves noisy rankings (ρ = +0.80 for H₂) while PEC actively reorders them (ρ = -0.22). Noisy expressibility, computed from density matrix simulations, strongly predicts unmitigated performance for H₃⁺ (Pearson r = +0.91, p = 0.01), but this metric is computationally intractable at scale. We demonstrate that zero-cost circuit topology metrics such as two-qubit gate count provide comparable or superior predictive power for PEC degradation (r = +0.96 for H₃⁺), while standard expressibility best predicts noisy and ZNE performance for H₂ (r = +0.74 and r = +0.77).