Scaling of errors in digitized counterdiabatic driving

We study errors caused by digitization of shortcuts to adiabaticity by counterdiabatic driving. We find possibility of error scaling $\mathcal{O}(M^{-2})$ with the number of time slices $M$, whereas worse error scaling $\mathcal{O}(M^{-1})$ is predicted in the conventional theory of the first-order Suzuki-Trotter decomposition. We point out this possibility by considering a state-dependent error bound and confirm emergence of this error scaling $\mathcal{O}(M^{-2})$ by numerical simulation. Moreover, we numerically show that intermediate error scaling can be observed in digitization of approximate counterdiabatic driving. These results reveal usefulness of digitized counterdiabatic driving from the viewpoints of both cost and performance.