Grover's Algorithm - Quantum Partial Search

Quantum Partial Search

A modification of Grover's algorithm called quantum partial search was described by Grover and Radhakrishnan in 2004. In partial search, one is not interested in finding the exact address of the target item, only the first few digits of the address. Equivalently, we can think of "chunking" the search space into blocks, and then asking "in which block is the target item?". In many applications, such a search yields enough information if the target address contains the information wanted. For instance, to use the example given by L.K. Grover, if one has a list of students organized by class rank, we may only be interested in whether a student is in the lower 25%, 25-50%, 50-70% or 75-100% percentile.

To describe partial search, we consider a database separated into blocks, each of size . Obviously, the partial search problem is easier. Consider the approach we would take classically - we pick one block at random, and then perform a normal search through the rest of the blocks (in set theory language, the compliment). If we don't find the target, then we know it's in the block we didn't search. The average number of iterations drops from to .

Grover's algorithm requires iterations. Partial search will be faster by a numerical factor which depends the number of blocks . Partial search uses global iterations and local iterations. The global Grover operator is designated and the local Grover operator is designated .

The global Grover operator acts acts on the blocks. Essentially, it is given as follows:

1. Perform standard Grover iterations on the entire database.
2. Perform local Grover iterations. A local Grover iteration is a direct sum of Grover iterations over each block.
3. Perform one standard Grover iteration

The optimal values of and are discussed in the paper by Grover and Radhakrishnan. One might also wonder what happens if one applies successive partial searches at different levels of "resolution". This idea was studied in detail by Korepin and Xu, who called it binary quantum search. They proved that it is not in fact any faster than performing a single partial search.