  As quantum computing hardware steadily increases in qubit count and quality,
one important question is how to allocate these resources to mitigate the
effects of hardware noise. In a transitional era between noisy small-scale and
fully fault-tolerant systems, we envisage a scenario in which we are only able
to error correct a small fraction of the qubits required to perform an
interesting computation. In this work, we develop concrete constructions of
logical operations on a joint system of a collection of noisy and a collection
of error-corrected logical qubits. Within this setting and under Pauli noise
assumptions, we provide analytical evidence that brick-layered circuits display
on average slower concentration to the "useless" uniform distribution with
increasing circuit depth compared to fully noisy circuits. We corroborate these
findings by numerical demonstration of slower decoherence with an increasing
fraction of error-corrected qubits under a noise model inspired by a real
device. We find that this advantage only comes when the number of
error-corrected qubits passes a specified threshold which depends on the number
of couplings between error-corrected and noisy registers.
