  We investigate the exact effect on ion trap quantum computation after field
quantization. First an exact expression of failure probability from field
quantization after many CNOT operations in Cirac-Zoller scheme is given. It is
proportional to operation number and the amplitude of $|1\rangle_x |0\rangle_y$
or $|1\rangle_x |1\rangle_y$ in initial state, and inverse proportional to mean
number of photons and amplitude of $|0\rangle_x |0\rangle_y$ or $|0\rangle_x
|1\rangle_y$ in initial state. Then we calculate the failure probability when
the limitation to mean number of photons in sideband transition is considered.
When the initial state is $|1\rangle_x |0\rangle_y$ or $|1\rangle_x
|1\rangle_y$, after about $10^2$ times of CNOT operations, failure probability
is no less than $10^{-2}$, while $10^{-2}$ is the known maximum threshold in
fault-tolerant quantum computation. Then when the initial state is $|1\rangle_x
|0\rangle_y$ or $|1\rangle_x |1\rangle_y$, the number of CNOT gates on the same
pair of physical qubits should be no more than $10^2$ in one error-correction
period, or else the computation cannot be implemented reliably. This conclusion
can help to determine the number of CNOT operations between coding and decoding
in one error-correction period in fault-tolerant quantum computation.
