Grover'S Search Algorithm With An Entangled Database State

Grover's oracle based unstructured search algorithm is often stated as "given a phone number in a directory, find the associated name." More formally, the problem can be stated as "given as input a unitary black box U-f for computing an unknown function f:{0,1}(n)->{0,1} find x=x(0) an element of {0,1}(n) such that f(x(0)) = 1, (and zero otherwise). The crucial role of the externally supplied oracle U-f (whose inner workings are unknown to the user) is to change the sign of the solution vertical bar x(0)>, while leaving all other states unaltered. Thus, U-f depends on the desired solution x(0). This paper examines an amplitude amplification algorithm in which the user encodes the directory (e. g. names and telephone numbers) into an entangled database state, which at a later time can be queried on one supplied component entry (e. g. a given phone number t(0)) to find the other associated unknown component (e. g. name x(0)). For N=2(n) names vertical bar x > with N associated phone numbers vertical bar t >, performing amplitude amplification on a subspace of size N of the total space of size N-2 produces the desired state vertical bar x(0)>vertical bar t(0)> in root N steps. We discuss how and why sequential (though not concurrent parallel) searches can be performed on multiple database states. Finally, we show how this procedure can be generalized to databases with more than two correlated lists (e.g. vertical bar x >vertical bar t >vertical bar s >vertical bar r > ... ).