  In a quantum computer any superposition of inputs evolves unitarily into the
corresponding superposition of outputs. It has been recently demonstrated that
such computers can dramatically speed up the task of finding factors of large
numbers -- a problem of great practical significance because of its
cryptographic applications. Instead of the nearly exponential ($\sim \exp
L^{1/3}$, for a number with $L$ digits) time required by the fastest classical
algorithm, the quantum algorithm gives factors in a time polynomial in $L$
($\sim L^2$). This enormous speed-up is possible in principle because quantum
computation can simultaneously follow all of the paths corresponding to the
distinct classical inputs, obtaining the solution as a result of coherent
quantum interference between the alternatives. Hence, a quantum computer is
sophisticated interference device, and it is essential for its quantum state to
remain coherent in the course of the operation. In this report we investigate
the effect of decoherence on the quantum factorization algorithm and establish
an upper bound on a ``quantum factorizable'' $L$ based on the decoherence
suffered per operational step.
