  We propose a scheme for scalable and robust quantum computing on
two-dimensional arrays of qubits with fixed longitudinal coupling. This opens
the possibility for bypassing the device complexity associated with tunable
couplers required in conventional quantum computing hardware. Our approach is
based on driving a subarray of qubits such that the total multi-qubit
Hamiltonian can be decomposed into a sum of commuting few-qubit blocks, and
then efficient optimization of the unitary evolution within each block. Each
driving pulse can implement a target gate on the driven qubits, and at the same
time implement identity gates on the neighbouring undriven qubits, cancelling
any unwanted evolution due to the constant qubit-qubit interaction. We show
that it is possible to realise a universal set of quantum gates with high
fidelity on the basis blocks, and by shifting the driving pattern one can
realise an arbitrary quantum circuit on the array. Allowing for imperfect
Hamiltonian characterisation, we use robust optimal control to obtain
fidelities around 99.99% despite 1% uncertainty in the qubit-qubit and
drive-qubit couplings, and a detuning uncertainty at 0.1% of the qubit-qubit
coupling strength. This robust feature is crucial for scaling up as parameter
uncertainty is significant in large devices.
