  We provide compelling evidence for the presence of quantum chaos in the
unitary part of Shor's factoring algorithm. In particular we analyze the
spectrum of this part after proper desymmetrization and show that the
fluctuations of the eigenangles as well as the distribution of the eigenvector
components follow the CUE ensemble of random matrices, of relevance to
quantized chaotic systems that violate time-reversal symmetry. However, as the
algorithm tracks the evolution of a single state, it is possible to employ
other operators, in particular it is possible that the generic quantum chaos
found above becomes of a nongeneric kind such as is found in the quantum cat
maps, and in toy models of the quantum bakers map.
