Editing Grover's algorithm (section)

===Quantum partial search===
A  modification of Grover's algorithm called quantum partial search was described by Grover and Radhakrishnan in 2004.<ref>{{cite arXiv
| eprint=quant-ph/0407122v4
| title=Is partial quantum search of a database any easier?
|first1=L. K. |last1=Grover |first2=J. |last2=Radhakrishnan | date=2005-02-07
}}</ref> 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%, 50–75% or 75–100% percentile.

To describe partial search, we consider a database separated into <math>K</math> blocks, each of size <math>b = N/K</math>. 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 complement). If we do not find the target, then we know it is in the block we did not search. The average number of iterations drops from <math>N/2</math> to <math>(N-b)/2</math>.

Grover's algorithm requires <math display="inline">\frac{\pi}{4}\sqrt{N}</math> iterations. Partial search will be faster by a numerical factor that depends on the number of blocks <math>K</math>. Partial search uses <math>n_1</math> global iterations and <math>n_2</math> local iterations. The global Grover operator is designated <math>G_1</math> and the local Grover operator is designated <math>G_2</math>.

The global Grover operator acts on the blocks. Essentially, it is given as follows:
#Perform <math>j_1</math> standard Grover iterations on the entire database.
#Perform <math>j_2</math> local Grover iterations. A local Grover iteration is a direct sum of Grover iterations over each block.
#Perform one standard Grover iteration.

The optimal values of <math>j_1</math> and <math>j_2</math> 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 [[Vladimir 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.