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We provide compelling evidence for the presence of quantum chaos in theunitary part of Shor's factoring algorithm. In particular we analyze thespectrum of this part after proper desymmetrization and show that thefluctuations of the eigenangles as well as the distribution of the eigenvectorcomponents follow the CUE ensemble of random matrices, of relevance toquantized chaotic systems that violate time-reversal symmetry. However, as thealgorithm tracks the evolution of a single state, it is possible to employother operators, in particular it is possible that the generic quantum chaosfound above becomes of a nongeneric kind such as is found in the quantum catmaps, and in toy models of the quantum bakers map.Comment: Title and paper modified to include interesting additional  possibilities. Principal results unaffected. Accepted for publication in  Phys. Rev. E as Rapid Com

Quantum chaos in the spectrum of operators used in Shor’s algorithm