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Fault-tolerant quantum computing demands many qubits with long lifetimes toconduct accurate quantum gate operations. However, external noise limits thecomputing time of physical qubits. Quantum error correction codes may extendsuch limits, but imperfect gate operations introduce errors to the correctionprocedure as well. The additional gate operations required due to the physicallayout of qubits exacerbate the situation. Here, we use density-matrixsimulations to investigate the performance change of logical qubits accordingto quantum error correction codes and qubit layouts and the expectedperformance of logical qubits with gate operation time and gate error rates.Considering current qubit technology, the small quantum error correction codesare chosen. Assuming 0.1% gate error probability, a logical qubit encoded by a5-qubit quantum error correction code is expected to have a fidelity 0.25higher than its physical counterpart.

Density-Matrix Simulation of Logical Qubit using 3-qubit Quantum Error Correction Code