  Measurement-induced phase transitions arise due to a competition between the
scrambling of quantum information in a many-body system and local measurements.
In this work we investigate these transitions in different classes of fast
scramblers, systems that scramble quantum information as quickly as is
conjectured to be possible -- on a timescale proportional to the logarithm of
the system size. In particular, we consider sets of deterministic sparse
couplings that naturally interpolate between local circuits that slowly
scramble information and highly nonlocal circuits that achieve the
fast-scrambling limit. We find that circuits featuring sparse nonlocal
interactions are able to withstand substantially higher rates of local
measurement than circuits with only local interactions, even at comparable gate
depths. We also study the quantum error-correcting codes that support the
volume-law entangled phase and find that our maximally nonlocal circuits yield
codes with nearly extensive contiguous code distance. Use of these sparse,
deterministic circuits opens pathways towards the design of noise-resilient
quantum circuits and error correcting codes in current and future quantum
devices with minimum gate numbers.
