  We investigate the problem of factorization of large numbers on a quantum
computer which we imagine to be realized within a linear ion trap. We derive
upper bounds on the size of the numbers that can be factorized on such a
quantum computer. These upper bounds are independent of the power of the
applied laser. We investigate two possible ways to implement qubits, in
metastable optical transitions and in Zeeman sublevels of a stable ground
state, and show that in both cases the numbers that can be factorized are not
large enough to be of practical interest. We also investigate the effect of
quantum error correction on our estimates and show that in realistic systems
the impact of quantum error correction is much smaller than expected. Again no
number of practical interest can be factorized.
