In this thesis, we present recent advances at the precision frontier of higher-order quantum chromodynamics (QCD) calculations. We consider massless two-loop five-point amplitudes, with a particular focus on diphoton-plus-jet production through gluon fusion. We build a library of infrared functions up to at most next-to-next-to-leading order (NNLO) in QCD, which can be used to validate amplitudes and construct counterterms in subtraction schemes at NNLO. We review progress in the novel use of machine learning technology to optimise the evaluation of amplitudes in hadron collider simulations. We present the full-colour virtual QCD corrections to diphoton-plus-jet production through gluon fusion, discussing the new techniques developed to calculate these non-planar two-loop amplitudes. We use these amplitudes to compute the next-to-leading QCD corrections to the differential cross sections of diphoton-plus-jet production through gluon fusion at the Large Hadron Collider. We also present the leading-colour double-virtual corrections to hadronic trijet production. All derived amplitudes are made available in a public implementation that is ready for further phenomenological application
