Fault-tolerant quantum computing demands many qubits with long lifetimes to conduct accurate quantum gate operations. However, external noise limits the computing time of physical qubits. Quantum error correction codes may extend such limits, but imperfect gate operations introduce errors to the correction procedure as well. The additional gate operations required due to the physical layout of qubits exacerbate the situation. Here, we use density-matrix simulations to investigate the performance change of logical qubits according to quantum error correction codes and qubit layouts and the expected performance of logical qubits with gate operation time and gate error rates. Considering current qubit technology, the small quantum error correction codes are chosen. Assuming 0.1% gate error probability, a logical qubit encoded by a 5-qubit quantum error correction code is expected to have a fidelity 0.25 higher than its physical counterpart.
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