  We use density matrix simulations to study the performance of three distance
three quantum error correcting codes in the context of the rare-earth-ion-doped
crystal (RE) platform for quantum computing. We analyze pseudothresholds for
these codes when parallel operations are not available, and examine the
behavior both with and without resting errors. In RE systems, resting errors
can be mitigated by extending the system's ground state coherence time. For the
codes we study, we find that if the ground state coherence time is roughly 100
times larger than the excited state coherence time, resting errors become small
enough to be negligible compared to other error sources. This leads us to the
conclusion that beneficial QEC could be achieved in the RE system with the
expected gate fidelities available in the NISQ regime. However, for codes using
more qubits and operations, a factor of more than 100 would be required.
Furthermore, we investigate how often QEC should be performed in a circuit. We
find that for early experiments in RE systems, the minimal $[\![5,1,3]\!]$
would be most suitable as it has a high threshold error and uses few qubits.
However, when more qubits are available the $[\![9,1,3]\!]$ surface code might
be a better option due to its higher circuit performance. Our findings are
important for steering experiments to an efficient path for realizing
beneficial quantum error correcting codes in early RE systems where resources
are limited.
