Expressibility, Noise, and Error Mitigation in VQE Ansatz Selection
For researchers using VQE on near-term quantum devices, this work shows that expressibility is an unreliable guide for ansatz selection under realistic noise and error mitigation.
The paper investigates whether expressibility predicts VQE performance under noise and error mitigation for H2 and H3+. It finds that error mitigation does not reliably restore expressibility's predictive power, and that simple circuit topology metrics can outperform expressibility in some cases.
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 $ρ\approx -0.1$ between ideal and noisy rankings), and identify a new result: ZNE largely preserves noisy rankings ($ρ= +0.80$ for $H_2$) while PEC actively reorders them ($ρ= -0.22$). 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 $r = +0.77$).