  Canyon landscapes in high dimension can be described as manifolds of small,
but extensive dimension, immersed in a higher dimensional ambient space and
characterized by a zero potential energy on the manifold. Here we consider the
problem of a quantum particle exploring a prototype of a high-dimensional
random canyon landscape. We characterize the thermal partition function and
show that around the point where the classical phase space has a satisfiability
transition so that zero potential energy canyons disappear, moderate quantum
fluctuations have a deleterious effect and induce glassy phases at temperature
where classical thermal fluctuations alone would thermalize the system.
Surprisingly we show that even when, classically, diffusion is expected to be
unbounded in space, the interplay between quantum fluctuations and the
randomness of the canyon landscape conspire to have a confining effect.
