  We elucidate the geometry of quantum adiabatic evolution. By minimizing the
deviation from adiabaticity we find a Riemannian metric tensor underlying
adiabatic evolution. Equipped with this tensor, we identify a unified geometric
description of quantum adiabatic evolution and quantum phase transitions, which
generalizes previous treatments to allow for degeneracy. The same structure is
relevant for applications in quantum information processing, including
adiabatic and holonomic quantum computing, where geodesics over the manifold of
control parameters correspond to paths which minimize errors. We illustrate
this geometric structure with examples, for which we explicitly find adiabatic
geodesics. By solving the geodesic equations in the vicinity of a quantum
critical point, we identify universal characteristics of optimal adiabatic
passage through a quantum phase transition. In particular, we show that in the
vicinity of a critical point describing a second order quantum phase
transition, the geodesic exhibits power-law scaling with an exponent given by
twice the inverse of the product of the spatial and scaling dimensions.
