  Ergodic theory provides a rigorous mathematical description of chaos in
classical dynamical systems, including a formal definition of the ergodic
hierarchy. How ergodic dynamics is reflected in the energy levels and
eigenstates of a quantum system is the central question of quantum chaos, but a
rigorous quantum notion of ergodicity remains elusive. Closely related to the
classical ergodic hierarchy is a less-known notion of cyclic approximate
periodic transformations [see, e.g., I. Cornfield, S. Fomin, and Y. Sinai,
Ergodic Theory (Springer-Verlag New York, 1982)], which maps any "ergodic"
dynamical system to a cyclic permutation on a circle and arguably represents
the most elementary form of ergodicity. This paper shows that cyclic ergodicity
generalizes to quantum dynamical systems, and provides a rigorous
observable-independent definition of quantum ergodicity. It implies the ability
to construct an orthonormal basis, where quantum dynamics transports any
initial basis vector to have a sufficiently large overlap with each of the
other basis vectors in a cyclic sequence. It is proven that the basis,
maximizing the overlap over all such quantum cyclic permutations, is obtained
via the discrete Fourier transform of the energy eigenstates. This relates
quantum cyclic ergodicity to energy level statistics. The level statistics of
Wigner-Dyson random matrices, usually associated with quantum chaos on
empirical grounds, is derived as a special case of this general relation. To
demonstrate generality, we prove that irrational flows on a 2D torus are
classical and quantum cyclic ergodic, with spectral rigidity distinct from
Wigner-Dyson. Finally, we motivate a quantum ergodic hierarchy of operators and
discuss connections to eigenstate thermalization. This work provides a general
framework for transplanting some rigorous concepts of ergodic theory to quantum
dynamical systems.
