Hierarchical quantum search

Database search has wide applications and is used as a subroutine in many important algorithms. In this paper we will consider a database with a single target item. Quantum algorithm [Grover] locates the target item faster than any classical algorithm. In addition to a full [Grover] search, it frequently occurs that one is looking for a group of items [a block] containing the target item, rather than the target item itself. This problem is known as partial search. As a generalization of the full search, partial search is of particular importance in practice. Partial search trades accuracy for speed, i.e. it works faster than a full search. There exists different versions of partial search. We will study the optimized version of the algorithm discovered by Grover and Radhakrishnan and call it GRK. GRK can be applied successively [in a sequence]. First the database is partitioned into blocks and GRK is applied to find the target block. Then this target block is partitioned into sub-blocks and GRK is used again to find the target sub-block. This procedure can be repeated if the database is large enough. [This sequence of GRK's is called a hierarchy.] Another possibility is to partition the database into sub-blocks directly and use GRK to find the target sub-block once. In this paper we will prove that the latter is faster [makes less queries to the oracle].