  When error correction becomes possible it will be necessary to dedicate a
large number of physical qubits to each logical qubit. Error correction allows
for deeper circuits to be run, but each additional physical qubit can
potentially contribute an exponential increase in computational space, so there
is a trade-off between using qubits for error correction or using them as noisy
qubits. In this work we look at the effects of using noisy qubits in
conjunction with noiseless qubits (an idealized model for error-corrected
qubits), which we call the "clean and dirty" setup. We employ analytical models
and numerical simulations to characterize this setup. Numerically we show the
appearance of Noise-Induced Barren Plateaus (NIBPs), i.e., an exponential
concentration of observables caused by noise, in an Ising model Hamiltonian
variational ansatz circuit. We observe this even if only a single qubit is
noisy and given a deep enough circuit, suggesting that NIBPs cannot be fully
overcome simply by error-correcting a subset of the qubits. On the positive
side, we find that for every noiseless qubit in the circuit, there is an
exponential suppression in concentration of gradient observables, showing the
benefit of partial error correction. Finally, our analytical models corroborate
these findings by showing that observables concentrate with a scaling in the
exponent related to the ratio of dirty-to-total qubits.
