There is a genuinely interesting complexity class called BQP/poly, which is pronounced something like “bounded-error quantum polynomial time with classical advice” (add some more syllables for a complete pronunciation).

The JVG algorithm is not a high quality example of this or really anything else. If you think of it as “classical advice”, then it fails, because the advice depends on the input and not just the size of the input. If you think of it as precomputation, it’s useless, because the precomputation involved already fully solves the discrete log problem. And the JVG paper doesn’t even explain how to run their circuit at respectable sizes without the sheer size of the circuit making the algorithm fail.

It’s a bit like saying that one could optimize Stockfish to run 1000x faster by giving it an endgame table covering all 16-or-fewer-piece-positions. Sure, maybe you could, but you also already solved chess by the time you finish making that table.