Quantum computers are very good for sampling distributions that we dont know how to sample using classical computers. For example if f is a Boolean function (from $\{-1,1\}^n$ to ${-1,1}$) that can be computed in polynomial time then with quantum computers we can efficiently sample according to the distribution described by the Fourier expansion of f. (We do not know how to do it with classical computers.) Can we use quantum computers to sample or approximately sample a random point  in a polyhedron described by a system of n inequalities in d variables?Moving from the inequalities to the points looks to me somewhat similar to a transform. Moreover, I would be happy to see a quantum algorithm even if you modify the distribution, e.g. consider product of Gaussian distribution described by the hyperplanes of the polyhedron or some other things.A few remarks: Dyer, Frieze and Kannan found a famous classical polynomial time algorithm to approximately sample and approximately compute the volume of a polyhedron. The algorithm is based on random walks and rapid mixing. So we want to find a different quantum algorithm for the same purpose. (OK, we can hope that a quantum algorithm may lead also to things in this context we do not know to do classically. But to start, all we want is a different algorithm, this must be possible.)Second, we dont even insist on approximately sampling the uniform distribution. We will be happy to approximately sample some other nice distribution which is roughly supported on our polyhedron. There is an argument by Santosh Vampala (and also by me in another context) leading from sampling to optimization: if you want to optimize f(x) sample to find a point y where f(x) is typical. Add the constraint {f(x)>= f(y)} and repeat.   
Approximately sampling from convex polyhedrons with quantum computers
quantum computing;cg.comp geom;optimization