  Quantum circuits implementing fault-tolerant quantum error correction (QEC)
for the three qubit bit-flip code and five-qubit code are studied. To describe
the effect of noise, we apply a model based on a generalized effective
Hamiltonian where the system-environment interactions are taken into account by
including stochastic fluctuating terms in the system Hamiltonian. This noise
model enables us to investigate the effect of noise in quantum circuits under
realistic device conditions and avoid strong assumptions such as maximal
parallelism and weak storage errors. Noise thresholds of the QEC codes are
calculated. In addition, the effects of imprecision in projective measurements,
collective bath, fault-tolerant repetition protocols, and level of parallelism
in circuit constructions on the threshold values are also studied with emphasis
on determining the optimal design for the fault-tolerant QEC circuit. These
results provide insights into the fault-tolerant QEC process as well as useful
information for designing the optimal fault-tolerant QEC circuit for particular
physical implementation of quantum computer.
