Expressibility, Noise, and Error Mitigation in VQE Ansatz Selection

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 $H_2$. We extend this investigation to cover both $H_2$ and $H_3^{+}$ 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 $H_2$ circuits and 4 of 6 $H_3^{+}$ circuits, while PEC actually increases error in 11 of 12 $H_2$ circuits and all 6 $H_3^{+}$ circuits. We reproduce and extend Saib et al.'s key finding that circuit rankings scramble under noise (Spearman $\rho \approx -0.1$ between ideal and noisy rankings), and identify a new result: ZNE largely preserves noisy rankings ($\rho =+0.80$ for $H_2$) while PEC actively reorders them ($\rho =-0.22$). Noisy expressibility, computed from density matrix simulations, strongly predicts unmitigated performance for $H_3^{+}$ (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 $H_3^{+}$), while standard expressibility best predicts noisy and ZNE performance for $H_2$ ($r=+0.74$ and $r=+0.77$).