Speedup of Iterated Quantum Search by Parallel Performance

Given a sequence f1(x1), f2(x1, x2), . . . , fk(x1, . . . , xk) of boolean functions, each fi takes the value 1 at a single point of the form x 0 1, x 0 2, . . . , xi , i " 1, 2, . . . , k. The length of all xi is n, N " 2n. It is shown how to find xk (k # 2) using kΠ ! N/4 ! 2 simultaneous evaluations of functions of the form fi, fi%1 with an error probability of order k/ ! N. This is ! 2 times faster than k sequential applications of the Grover algorithm for quantum search. Evolutions of amplitudes in parallel quantum computations are approximated by systems of linear differential equations. Some advantage of simultaneous evaluations of all f1, . . . fk are discussed.