Symanzik improvement of the gradient flow in lattice gauge theories

We apply the Symanzik improvement programme to the $$4+1$$-dimensional local re-formulation of the gradient flow in pure SU(N) lattice gauge theories. We show that the classical nature of the flow equation allows one to eliminate all cutoff effects at $$\mathcal O(a^2)$$, which originate either from the discretised gradient flow equation or from the gradient flow observable. All the remaining $$\mathcal O(a^2)$$ effects can be understood in terms of local counterterms at the zero flow-time boundary. We classify these counterterms and provide a complete set as required for on-shell improvement. Compared to the 4-dimensional pure gauge theory only a single additional counterterm is required, which corresponds to a modified initial condition for the flow equation. A consistency test in perturbation theory is passed and allows one to determine all counterterm coefficients to lowest non-trivial order in the coupling