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It is pointed out that an exactly solvable permutation operator, viewed asthe quantization of cyclic shifts, is useful in constructing a basis in whichto study the quantum baker's map, a paradigm system of quantum chaos. In thebasis of this operator the eigenfunctions of the quantum baker's map arecompressed by factors of around five or more. We show explicitly its connectionto an operator that is closely related to the usual quantum baker's map. Thispermutation operator has interesting connections to the art of shuffling cardsas well as to the quantum factoring algorithm of Shor via the quantum orderfinding one. Hence we point out that this well-known quantum algorithm makescrucial use of a quantum chaotic operator, or at least one that is close to thequantization of the left-shift, a closeness that we also explorequantitatively.Comment: 12 pgs. Substantially elaborated version, including a new route to  the quantum bakers map. To appear in J. Phys.

Shuffling cards, factoring numbers and the quantum baker's map