Results and techniques for higher order calculations within the gradient-flow formalism

We describe in detail the implementation of a systematic perturbative approach to observables in the QCD gradient-flow formalism. This includes a collection of all relevant Feynman rules of the five-dimensional field theory and the composite operators considered in this paper. Tools from standard perturbative calculations are used to obtain Green's functions at finite flow time t at higher orders in perturbation theory. The three-loop results for the quark condensate at finite t and the conversion factor for the "ringed" quark fields to the $\overline{MS}$ scheme are presented as applications. We also re-evaluate an earlier result for the three-loop gluon condensate, improving on its accuracy.