  We propose a quantum heuristic algorithm to solve a traveling salesman
problem by generalizing Grover search. Sufficient conditions are derived to
greatly enhance the probability of finding the tours with extremal costs,
reaching almost to unity and they are shown characterized by statistical
properties of tour costs. In particular for a Gaussian distribution of the
tours along the cost we show that the quantum algorithm exhibits the quadratic
speedup of its classical counterpart, similarly to Grover search.
