Consider the following computational task: We want to sample a 3-SAT formula of $n$ variables (a variant: $n$ variables $m$ clauses) with respect to the uniform probability distribution, conditioned on the formula being satisfiable:
Q1: Can this be achieved efficiently by a classical computer (with random bits)?
Q2: Can this be achieved efficiently by a quantum computer?
I am also interested in the following two variants:
V2: You sample all the formulas w.r.t. a probability distribution that gives satisfiable formulas twice the weight of unsatisfiable formulas.
V3: you sample where the weight is the number of satisfying assignments (Here we care only about Q2).
Update: Colins' answer demonstrates a simple algorithm for V3. (I was wrong in assuming that this is classically difficult.) Let me mention another variant of all the three questions:
You specify in advance $m$ clauses and you need to sample random satisfiable subsets of the input clauses.