Approximately sampling from convex polyhedrons with quantum computers
As the post acknowledges, the existence of a classical polynomial-time algorithm to estimate the volume of a convex polytope is a game-changer.  A quantum algorithm is much less likely to be interesting unless it is competitive with classical algorithms.   After all, without that criterion, any classical algorithm could simply be called a quantum algorithm instead.

That said, there is still room for a polynomial speed-up, and the main, known viewpoint for that type of speedup is a quantum walk, especially considering that the classical acceleration in this case is based on a good random walk.   (Indeed, any quantum algorithm can be viewed as a quantum walk, but for some algorithms this is not necessarily enlightening.)   Various papers in the QC literature have pointed out that the algorithms to estimate the volume of a convex polytope use random walks, and that there could be an acceleration from a quantum walk.  So, it appears that researchers know this suggestion, but that no one has tried to work out what polynomial acceleration you might get for this problem.   You might not get anything if the best classical algorithm has some sort of spoiler, such as a separate stage which is not a random walk with comparable running time.

Here is a collection of papers that all mention the basic idea in passing; again, Google Scholar seems to suggest that no one has gone further.


arXiv:quant-ph/0104137 - Quantum Walks on the Hypercube
arXiv:quant-ph/0205083 - Quantum random walks hit exponentially faster
arXiv:quant-ph/0301182 - Decoherence in Discrete quantum walks
arXiv:quant-ph/0304204 - Controlling discrete quantum walks: coins and intitial states
arXiv:quant-ph/0411065 - Quantum walk on a line with two entangled particles
arXiv:quant-ph/0504042 - Entanglement in coined quantum walks on regular graphs
arXiv:quant-ph/0609204 - Quantum speedup of classical mixing processes
arXiv:0804.4259 - Speed-up via quantum sampling
A random walk approach to quantum algorithms
Discrete quantum walk for solving nonlinear equations over finite fields


The other side of the classical algorithms to estimate the volume of a convex polytope is linear programming.   I don't know there has been any progress finding a quantum acceleration for that.  It seems difficult to avoid a stage of linear programming in order to put the convex polytope in a favorable position for sampling.