  As superconducting quantum circuits scale to larger sizes, the problem of
frequency crowding proves a formidable task. Here we present a solution for
this problem in fixed-frequency qubit architectures. By systematically
adjusting qubit frequencies post-fabrication, we show a nearly ten-fold
improvement in the precision of setting qubit frequencies. To assess
scalability, we identify the types of 'frequency collisions' that will impair a
transmon qubit and cross-resonance gate architecture. Using statistical
modeling, we compute the probability of evading all such conditions, as a
function of qubit frequency precision. We find that without post-fabrication
tuning, the probability of finding a workable lattice quickly approaches 0.
However with the demonstrated precisions it is possible to find collision-free
lattices with favorable yield. These techniques and models are currently
employed in available quantum systems and will be indispensable as systems
continue to scale to larger sizes.
