  The Gradient Ascent Pulse Engineering (GRAPE) is a celebrated control
algorithm with excellent converging rates, owing to a piece-wise-constant
ansatz for the control function that allows for cheap objective gradients.
However, the computational effort involved in the exact simulation of quantum
dynamics quickly becomes a bottleneck limiting the control of large systems. In
this paper, we propose a modified version of GRAPE that uses Krylov
approximations to deal efficiently with high-dimensional state spaces. Even
though the number of parameters required by an arbitrary control task scales
linearly with the dimension of the system, we find a constant elementary
computational effort (the effort per parameter). Since the elementary effort of
GRAPE is super-quadratic, this speed up allows us to reach dimensions far
beyond. The performance of the K-GRAPE algorithm is benchmarked in the
paradigmatic XXZ spin-chain model.
