  We address the difference between integrable and chaotic motion in quantum
theory as manifested by the complexity of the corresponding evolution
operators. Complexity is understood here as the shortest geodesic distance
between the time-dependent evolution operator and the origin within the group
of unitaries. (An appropriate `complexity metric' must be used that takes into
account the relative difficulty of performing `nonlocal' operations that act on
many degrees of freedom at once.) While simply formulated and geometrically
attractive, this notion of complexity is numerically intractable save for toy
models with Hilbert spaces of very low dimensions. To bypass this difficulty,
we trade the exact definition in terms of geodesics for an upper bound on
complexity, obtained by minimizing the distance over an explicitly prescribed
infinite set of curves, rather than over all possible curves. Identifying this
upper bound turns out equivalent to the closest vector problem (CVP) previously
studied in integer optimization theory, in particular, in relation to
lattice-based cryptography. Effective approximate algorithms are hence provided
by the existing mathematical considerations, and they can be utilized in our
analysis of the upper bounds on quantum evolution complexity. The resulting
algorithmically implemented complexity bound systematically assigns lower
values to integrable than to chaotic systems, as we demonstrate by explicit
numerical work for Hilbert spaces of dimensions up to ~10^4.
