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Markovian regime decoherence effects in quantum computers are studied in terms of the fidelity for the situation where the number of qubits N becomes large. A standard model involving three-state lambda system ionic qubits is considered, with qubits localised around well-separated positions via trapping potentials. The centre of mass vibrations of the qubits act as a reservoir. Coherent one and two qubit gating processes are controlled by time dependent localised classical EM fields that address specific qubits, the two qubit gating processes being facilitated by a cavity mode ancilla, which permits state interchange between qubits. With suitably chosen parameters, the decoherence time can be made essentially independent of N. © 2004 American Institute of Physics [ABSTRACT FROM AUTHOR]