  Optimal control can be used to significantly improve multi-qubit gates in
quantum information processing hardware architectures based on superconducting
circuit quantum electrodynamics. We apply this approach not only to dispersive
gates of two qubits inside a cavity, but, more generally, to architectures
based on two-dimensional arrays of cavities and qubits. For high-fidelity gate
operations, simultaneous evolutions of controls and couplings in the two
coupling dimensions of cavity grids are shown to be significantly faster than
conventional sequential implementations. Even under experimentally realistic
conditions speedups by a factor of three can be gained. The methods immediately
scale to large grids and indirect gates between arbitrary pairs of qubits on
the grid. They are anticipated to be paradigmatic for 2D arrays and lattices of
controllable qubits.
