My two cents too: this can work only for martingale processes. I don't know G2++, but it seems that it is not a martingale process ? Could this explain the 3% diff ?
Considering Sobol or Halton, I think we can provide sequences that outperforms them.
There ought to be a martingale there somewhere. E.g., pick the zero coupon bond with the latest maturity date that you care about and look at the ratio of other prices to that. That will be a martingale in the relevant forward measure, independent of any details of the model dynamics, and this measure is normally fairly convenient to work with.
I was just singling out Halton and Sobol because they’re probably the best known. And when used with some care they’re not usually that bad, although I agree that you can undoubtedly do better.
I agree to say that in every stochastic process there is a martingale part and a drift (hyperbolic) part. However, and I think that it is the question of the author of this post, it is not clear to me how to separate both parts just from price observations. Indeed, the author provides a moment matching method, that should be able to match any price of bonds, being martingale in the relevant forward measure, exactly what you said. His method should work for that purpose, but might fail to match swaption prices.
Passing by, I found this alternative reference
expliciting the calibration with G2++ from bonds or swaptions prices (Brigo is almost surely a good one too).
Concerning Halton and Sobol I agree, they are known and provide already good results (for low dimensions, take care). I was simply advertizing that we can propose better alternatives if more accurate results are needed. In the same spirit, we can also provide a general method to calibrate from any set of prices of derivatives, be they bonds or swaptions or whatever. The calibrated process is computed as "as close as possible" to the initial process (here a G2++) while constrained to reproduce this set of prices. Both methods works, but they require a more sophisticated approach.