Expressibility, Noise, and Error Mitigation in VQE Ansatz Selection

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 $ρapprox -0.1$ 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 $r = +0.77$.