Quantum Search with Variable Times
Andris Ambainis · Theory of Computing Systems · 2009
Since Grover’s seminal work, quantum search has been studied in great detail. In the usual search problem, we have a collection of n items x 1,…,x n and we would like to find i:x i =1. We consider a new variant of this problem in which evaluating x i for different i may take a different number of time steps. Let t i be the number of time steps required to evaluate x i . If the numbers t i are known in advance, we give an algorithm that solves the problem in $O(\sqrt{t_{1}^{2}+t_{2}^{2}+\ldots+t_{n}^{2}})$ steps. This is optimal, as we also show a matching lower bound. The case, when t i are not known in advance, can be solved with a polylogarithmic overhead. We also give an application of our new search algorithm to computing read-once functions.