  In quantum computation, quantum coherence must be maintained during gate
operation. However, in physical implementations, various couplings with the
environment are unavoidable and can lead to a dephasing of a quantum
bit(qubit). The background charge fluctuations are an important dephasing
process, especially in a charge qubit system. We examined the dephasing rate of
a qubit due to random telegraph noise. Solving stochastic differential
equations, we obtained the dephasing rate of a qubit constructed of a
coupled-dot system; we applied our results to the charge Josephson qubit
system. We examined the dephasing rates due to two types of couplings between
the coupled-dot system and the background charge, namely, fluctuation in the
tunnel coupling constant and fluctuation in the asymmetric bias. For a strong
coupling condition, the dephasing rate was inversely proportional to the time
constant of the telegraph noise. When there is fluctuation in the tunnel
coupling constant, Gaussian decay occurs in the initial regime. We also
examined the rate of dephasing due to many impurity sites. For a weak coupling
condition with fluctuation in the asymmetric bias, the obtained dephasing rate
coincided with that obtained by the perturbation method using the spectral
weight of a boson thermal bath, which is proportional to the inverse of the
frequency.
