  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.
