  Shor's factoring algorithm, believed to provide an exponential speedup over
classical computation, relies on finding the period of an exactly periodic
quantum modular multiplication operator. This exact periodicity is the hallmark
of an integrable system, which is paradoxical from the viewpoint of quantum
chaos, given that the classical limit of the modular multiplication operator is
a highly chaotic system that occupies the "maximally random" Bernoulli level of
the classical ergodic hierarchy. In this work, we approach this apparent
paradox from a quantum dynamical systems viewpoint, and consider whether
signatures of ergodicity and chaos may indeed be encoded in such an
"integrable" quantization of a chaotic system. We show that Shor's modular
multiplication operator, in specific cases, can be written as a superposition
of quantized A-baker's maps exhibiting more typical signatures of quantum chaos
and ergodicity. This work suggests that the integrability of Shor's modular
multiplication operator may stem from the interference of other "chaotic"
quantizations of the same family of maps, and paves the way for deeper studies
on the interplay of integrability, ergodicity and chaos in and via quantum
algorithms.
