Quantum error correction in the noisy intermediate-scale quantum regime for sequential quantum computing

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