NP-complete problems and Quantum Search

Consider SAT, the prototypical example of an NP-complete problem. An instance of this problem consists of a Boolean function f (x1, . . . ,xn) = c1 ∧ . . .∧ cm; the SAT problem asks you to determine whether there exists a satisfying assignment—that is, an input (a1, . . . ,an) such that f (a1, . . . ,an) = 1. UNIQUE-SAT is a variant of SAT that poses the same problem with the restriction that f must have zero or one satisfying assignments, but no more. As it turns out, there is a randomized reduction from SAT to UNIQUE-SAT; thus, the two problems are equally hard. We’ll use the black box model when considering this problem. In this model, we know that either f ≡ 0 or there exists exactly one a such that f (a) = 1, where a is chosen uniformly at random. That is, f is treated as a black box; we can make queries to f , but we have no access to the Boolean formula itself. Equivalently we can represent f by a table of N = 2n entries where either none or exactly one entry is 1. Ideally we want a quantum algorithm that solves this problem in time O(poly(n)) = O(poly− log(N)).