I've recently experimented with Quasi-random numbers in monte-carlo applications. Is there a way to construct a quasi-random shuffle? By that I mean can I take a sequence $Q$ and shuffle it to produce a sequence $S$ where $\frac{1}{j} \sum_{i=1}^{j} f(S_i)$ converges to the whole-sequence mean faster than a random shuffle.
Apologies in advance if I got the notation wrong up there, I'm not a mathematician.
You can follow the approach presented in the paper Quasi-Random resamplings, with applications to rule-samplng, cross-validation and (su-)bagging. Additionally if feasible I would suggest keeping track of previous draws, as repetitions might arise before the whole set of permutations is exhausted.
Depending on the form of your function (e.g. if there is a nonuniform relationship among coordinates) better methods could also be devised.