We analyze a randomly perturbed quantum version of the baker’s transformation, a prototype of an area-conserving chaotic map. By simulating the perturbed evolution, we estimate the information needed to follow a perturbed Hilbert-space vector in time. We find that the Landauer erasure cost associated with this grows very rapidly and becomes larger than the maximum statistical entropy given by the logarithm of the dimension of Hilbert space. The quantum baker’s map displays a hypersensitivity to perturbations analogous to behavior found in the classical case. This hypersensitivity characterizes ‘‘quantum chaos’’in a way that is relevant to statistical physics.