Resource-aware error mitigation for variational quantum eigensolvers: coupling zero-noise extrapolation, shot allocation and adaptive circuit growth on noisy hardware models

Variational quantum eigensolvers (VQE) on noisy intermediate-scale quantum (NISQ) hardware are limited by three coupled resources: gate noise that biases every energy evaluation, the finite shot budget that sets the statistical precision, and the circuit depth that both the ansatz and any noise-amplifying mitigation scheme consume. Zero-noise extrapolation (ZNE), variance-aware shot allocation and adaptive ansatz construction are each established in isolation, yet they interact: extrapolation amplifies variance, allocation changes the noise floor against which an adaptive method decides whether to grow, and growth changes the depth that extrapolation must fold. This paper presents RAM-VQE, a resource-aware mitigation framework in which the three mechanisms share one shot ledger and one noise model: adaptive operator selection is screened on the mitigated energy landscape with a significance test and geometric precision escalation, shots are allocated jointly across noise-scale factors and commuting measurement groups by a Neyman rule, and extrapolation uses a depolarizing-channel (exponential) model inside the loop and for the final estimate. Two propositions give the allocation gain and the extrapolation variance amplification in closed form. The framework is evaluated on a gate-level density-matrix model with per-gate depolarizing and readout errors, validated against PennyLane to 10−12 Ha, for H2 (4 qubits), stretched LiH (6 qubits) and a 4-site XXZ chain, against seven ablation arms over six seeds, with noise-rate and budget sweeps. At a two-qubit error rate of 5 × 10−3 the root-mean-square error relative to readout-mitigated VQE falls from 88.5 to 3.12 mHa (H2) and from 82.9 to 5.22 mHa (LiH, with 48.0 rather than 64.0 two-qubit gates), but at 8.65 times the shot cost of applying the same mitigation to a fixed ansatz, whose error is 6.83 mHa; on the spin chain the adaptive pool is ansatz-limited and the fixed ansatz is better. Three negative findings are reported: adaptive growth screened on raw noisy gains stalls at the reference state, in-loop extrapolation without shot allocation degrades the optimisation, and data-driven selection among extrapolation models from three scale factors is unreliable. Code, seeds and result logs are released.