Zeeman--Doppler imaging of early-type stars. Testing Stokes parameter inversions with theoretical magnetohydrodynamic models

Zeeman--Doppler imaging (ZDI) is widely used to reconstruct magnetic fields in early-type stars from time-series spectropolarimetric observations, yet its accuracy has not been systematically validated against physically self-consistent, complex magnetic field models. We aim to assess the reliability and limitations of modern ZDI techniques applied to fossil fields of early-type stars. We generated synthetic spectropolarimetric data from three-dimensional magnetohydrodynamic (MHD) simulations of fossil magnetic fields. The resulting Stokes profiles -- individual lines and least-squares deconvolved (LSD) profiles -- were inverted with contemporary ZDI codes using Stokes IQUV and IV data. We compared the reconstructed maps with the input MHD surface fields and quantified deviations in field strength, vector orientation, and harmonic energy decomposition. For Stokes IQUV inversions, ZDI recovers field vector components and strength with better than 10% accuracy, while Stokes IV inversions degrade to 15--20% for the most complex geometries. The field inclination is recovered to within 10 with errors increasing to 15--20 lines. Including linear polarisation offers a clear advantage for complex geometries. Imposing a current-free poloidal constraint, which the input field satisfies, only helps where the unconstrained Stokes IV inversion performs poorly. Cross-talk between magnetic and abundance reconstructions is weak, with spurious inhomogeneities at or below 0.1 dex on average. in the same cases. Full Stokes inversions reproduce poloidal and toroidal energies to within a few percentage points. LSD-based inversions yield accuracy comparable to, and sometimes better than, that achieved with ZDI using individual Fe ii Modern ZDI robustly reconstructs fossil magnetic fields in early-type stars, including their smaller-scale non-dipolar features, particularly when full Stokes information is available. Stokes IV-only inversions, however, can introduce measurable artefacts for complex topologies. ZDI reliably recovers the harmonic content of global fields, successfully disentangling poloidal and toroidal contributions. These results establish quantitative benchmarks for modern ZDI applied to early-type magnetic stars.