Next-to-next-to-leading order event generation for $t\bar{t}H$ production with approximate two-loop amplitude

We study Higgs-boson production in association with a top-quark pair ($t\bar{t}H$) at hadron colliders and present the first matching of next-to-next-to-leading order (NNLO) QCD corrections to parton showers using the MiNNLOPS method. For the two-loop amplitude, we employ two established approximations, based on the soft Higgs-boson and high-energy limits, respectively. For the first time, we also construct the latter in full colour and propose a pointwise combination of the two approximations across phase space. By assigning a conservative uncertainty estimate, which remains well below the perturbative uncertainties, we ensure robust and reliable differential predictions, explicitly validated at the one-loop level. Apart from the two-loop amplitude, all remaining ingredients of the MiNNLOPS calculation are included exactly. After thorough validation, we present a series of phenomenological results illustrating the impact of NNLO corrections and parton-shower effects. We consider fiducial predictions for the Higgs-boson decay into photons and include off-shell top-quark decays with tree-level spin correlations in both the dilepton and semileptonic channels. Our $t\bar{t}H$ MiNNLOPS generator is publicly available within the POWHEG framework.