  We propose a model based on a generalized effective Hamiltonian for studying
the effect of noise in quantum computations. The system-environment
interactions are taken into account by including stochastic fluctuating terms
in the system Hamiltonian. Treating these fluctuations as Gaussian Markov
processes with zero mean and delta function correlation times, we derive an
exact equation of motion describing the dissipative dynamics for a system of n
qubits. We then apply this model to study the effect of noise on the quantum
teleportation and a generic quantum controlled-NOT (CNOT) gate. For the quantum
CNOT gate, we study the effect of noise on a set of one- and two-qubit quantum
gates, and show that the results can be assembled together to investigate the
quality of a quantum CNOT gate operation. We compute the averaged gate fidelity
and gate purity for the quantum CNOT gate, and investigate phase, bit-flip, and
flip-flop errors during the CNOT gate operation. The effects of direct
inter-qubit coupling and fluctuations on the control fields are also studied.
We discuss the limitations and possible extensions of this model. In sum, we
demonstrate a simple model that enables us to investigate the effect of noise
in arbitrary quantum circuits under realistic device conditions.
