  Performing efficient quantum computer tuneup and calibration is essential for
growth in system complexity. In this work we explore the link between
facilitating such capabilities and the underlying architecture of the physical
hardware. We focus on the specific challenge of measuring (``mapping'')
spatially inhomogeneous quasi-static calibration errors using spectator qubits
dedicated to the task of sensing and calibration. We introduce a novel
architectural concept for such spectator qubits: arranging them spatially
according to prescriptions from optimal 2D approximation theory. We show that
this insight allows for efficient reconstruction of inhomogeneities in qubit
calibration, focusing on the specific example of frequency errors which may
arise from fabrication variances or ambient magnetic fields. Our results
demonstrate that optimal interpolation techniques display near optimal
error-scaling in cases where the measured characteristic (here the qubit
frequency) varies smoothly, and we probe the limits of these benefits as a
function of measurement uncertainty. For more complex spatial variations, we
demonstrate that the NMQA formalism for adaptive measurement and noise
filtering outperforms optimal interpolation techniques in isolation, and
crucially, can be combined with insights from optimal interpolation theory to
produce a general purpose protocol.
