Ambanini's algorithm will almost certainly never be used practically. It relies on Grover search to achieve its speedup, which has been basically shown to not be practical in the foreseeable future (see here for example. Held-Karp isn't used in practice either, since the exponential complexity is detrimental very quickly, and instead heuristics are used (this usually for example what popular software like Gurobi does). So extremely unlikely that TSP will be something quantum will help us with