The first task is to pick off the knights: 1.Qd1+ Rc1 2.Qd3+ Rc2 (Bc2 allows Qc3) 3.Qxf1+ Rc1 4.Qd3+ Rc2 5.Qd1+ Rc1 6.Qd2! Rc2 7.Qe1+ Rc1 8.Qe4+ Rc2 9.Qxh1+ Rc1.
Now we need a spot of delicacy: 10.Qe4+ Rc2 11.Qe1+ Rc1 12.Qd2 Rc2 13.Qd1+ Rc1 14.Qd3+ Rc2 15.Qe4! The same position as at move nine, but it's Black to play. Since 15...Kc1 allows mate on e1, and 15...Bb3 loses to 16.Qe1+ Rc1 17.Qd2 Rc2 18.Qd1+ Rc1 19.Qxb3, he has to move a pawn.
So 15...h2 16.Qe1+ Rc1 17.Qd2 Rc2 18.Qd1+ Rc1 19.Qd3+ Rc2 20.Qe4 and Black must move another pawn. After 20...h1=Q 21.Qxh1+ Rc1 22.Qe4+ Rc2 23.Qd2, the white queen gets back to e4 on move 27, forcing another pawn move. Black has, in all, 15 non-promoting pawn moves (including a6), each answered by a series of five checks, and four promotions on h1, answered by seven checks. So after 123.Qe4! Black has left only a pawn on h2. The solution then ends 123...h1=Q 124.Qxh1+ Rc1 125.Qh7+! Rc2 126.Qe4 Bb3 (he has nothing else) 127.Qe1+ Rc1 128.Qd2 Rc2 129.Qd1+ Rc1 130.Qxb3 mate! Bravo.