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Quantum error correction in the NISQ regime for sequential quantum computing

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.