In models of classical computation, searching an unsorted database cannot be done in less than linear time (so merely searching through every item is optimal). Grover's algorithm illustrates that in the quantum model searching can be done faster than this; in fact its time complexity O(N1/2) is asymptotically the fastest possible for searching an unsorted database in the quantum model. It provides a quadratic speedup, unlike other quantum algorithms, which may provide exponential speedup over their classical counterparts. However, even quadratic speedup is considerable when N is large.
Like many quantum algorithms, Grover's algorithm is probabilistic in the sense that it gives the correct answer with high probability. The probability of failure can be decreased by repeating the algorithm. (An example of a deterministic quantum algorithm is the Deutsch-Jozsa algorithm, which always produces the correct answer.)
Although the purpose of Grover's algorithm is usually described as "searching a database", it may be more accurate to describe it as "inverting a function". Roughly speaking, if we have a function y=f(x) that can be evaluated on a quantum computer, this algorithm allows us to calculate x when given y. Inverting a function is related to the searching of a database because we could come up with a function that produces a particular value of y if x matches a desired entry in a database, and another value of y for other values of x.
Grover's algorithm can also be used for estimating the mean and median of a set of numbers, and for solving the Collision problem. The algorithm can be further optimized if there is more than one matching entry and the number of matches is known beforehand.
Consider an unsorted database with N entries. The algorithm requires an N-dimensional state space H, which can be supplied by n=log2 N qubits. Consider the problem of determining the index of the database entry which satisfies some search criterion. Let f be the function which maps database entries to 0 or 1, where f(ω)=1 if and only if ω satisfies the search criterion. We are provided with (quantum black box) access to a subroutine in the form of a unitary operator, Uω, which acts as:
Our goal is to identify the index .
Algorithm steps 
The steps of Grover's algorithm are given as follows. Let denote the uniform superposition over all states
Then the operator
is known as the Grover diffusion operator.
Here is the algorithm:
- Initialize the system to the state
- Perform the following "Grover iteration" r(N) times. The function r(N), which is asymptotically O(N½), is described below.
- Apply the operator .
- Apply the operator .
- Perform the measurement Ω. The measurement result will be λω with probability approaching 1 for N≫1. From λω, ω may be obtained.
The first iteration 
A preliminary observation, in parallel with our definition
is that Uω can be expressed in an alternate way:
To prove this it suffices to check how Uω acts on basis states:
- for all .
The following computations show what happens in the first iteration:
After application of the two operators ( and ), the amplitude of the searched-for element has increased from to .
Description of 
Grover's algorithm requires a "quantum oracle" operator which can recognize solutions to the search problem and give them a negative sign. In order to keep the search algorithm general, we will leave the inner workings of the oracle as a black box, but will explain how the sign is flipped. The oracle contains a function which returns if is a solution to the search problem and otherwise. The oracle is a unitary operator which operates on two qubits, the index qubit and the oracle qubit :
As usual, denotes addition modulo 2. The operation flips the oracle qubit if and leaves it alone otherwise. In Grover's algorithm we want to flip the sign of the state if it labels a solution. This is achieved by setting the oracle qubit in the state , which is flipped to if is a solution:
We regard as flipped, thus the oracle qubit is not changed, so by convention the oracle qubits are usually not mentioned in the specification of Grover's algorithm. Thus the operation of the oracle is simply written as:
Geometric proof of correctness 
Consider the plane spanned by and , where is a ket in the subspace perpendicular to . We will consider the first iteration, acting on the initial ket . Since is one of the basis vectors in the overlap is
In geometric terms, the angle between and is given by:
The operator is a reflection at the hyperplane orthogonal to for vectors in the plane spanned by and ; i.e. it acts as a reflection across . The operator is a reflection through . Therefore, the state vector remains in the plane spanned by and after each application of the operators and , and it is straightforward to check that the operator of each Grover iteration step rotates the state vector by an angle of .
We need to stop when the state vector passes close to ; after this, subsequent iterations rotate the state vector away from , reducing the probability of obtaining the correct answer. The exact probability of measuring the correct answer is:
where r is the (integer) number of Grover iterations. The earliest time that we get a near-optimal measurement is therefore .
Algebraic proof of correctness 
To complete the algebraic analysis we need to find out what happens when we repeatedly apply . A natural way to do this is by eigenvalue analysis of a matrix. Notice that during the entire computation, the state of the algorithm is a linear combination of and . We can write the action of and in the space spanned by as:
So in the basis (which is neither orthogonal nor a basis of the whole space) the action of applying followed by is given by the matrix
This matrix happens to have a very convenient Jordan form. If we define , it is
It follows that rth power of the matrix (corresponding to r iterations) is
Using this form we can use trigonometric identities to compute the probability of observing ω after r iterations mentioned in the previous section,
Alternatively, one might reasonably imagine that a near-optimal time to distinguish would be when the angles 2rt and -2rt are as far apart as possible, which corresponds to or . Then the system is in state
A short calculation now shows that the observation yields the correct answer ω with error O(1/N).
Extension to space with multiple targets 
If, instead of 1 matching entry, there are k matching entries, the same algorithm works but the number of iterations must be π(N/k)1/2/4 instead of πN1/2/4. There are several ways to handle the case if k is unknown. For example, one could run Grover's algorithm several times, with
iterations. For any k, one of the iterations will find a matching entry with a sufficiently high probability. The total number of iterations is at most
which is still O(N1/2). It can be shown that this can be improved. If the number of marked items is k, where k is unknown, there is an algorithm that finds the solution in queries. This fact is used in order to solve the collision problem.,
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 complement). 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 on 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 on the blocks. Essentially, it is given as follows:
- Perform standard Grover iterations on the entire database.
- Perform 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 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.
It is known that Grover's algorithm is optimal. That is, any algorithm that accesses the database only by using the operator Uω must apply Uω at least as many times as Grover's algorithm. This result is important in understanding the limits of quantum computation. If the Grover's search problem was solvable with logc N applications of Uω, that would imply that NP is contained in BQP, by transforming problems in NP into Grover-type search problems. The optimality of Grover's algorithm suggests (but does not prove) that NP is not contained in BQP.
The number of iterations for k matching entries, π(N/k)1/2/4, is also optimal.
See also 
- Bennett C.H., Bernstein E., Brassard G., Vazirani U., The strengths and weaknesses of quantum computation. SIAM Journal on Computing 26(5): 1510–1523 (1997). Shows the optimality of Grover's algorithm.
- Michel Boyer; Gilles Brassard; Peter Høyer; Alain Tapp (1998), Fortsch. Phys. 46: 493–506, arXiv:quant-ph/9605034
- Andris Ambainis (2004), "Quantum search algorithms", SIGACT News 35 (2): 22–35, arXiv:quant-ph/0504012
- L.K. Grover and J. Radhakrishnan,Is partial quantum search of a database any easier?. quant-ph/0407122
- Grover L.K.: A fast quantum mechanical algorithm for database search, Proceedings, 28th Annual ACM Symposium on the Theory of Computing, (May 1996) p. 212
- Grover L.K.: From Schrödinger's equation to quantum search algorithm, American Journal of Physics, 69(7): 769-777, 2001. Pedagogical review of the algorithm and its history.
- Nielsen, M.A. and Chuang, I.L. Quantum computation and quantum information. Cambridge University Press, 2000. Chapter 6.
- What's a Quantum Phone Book?, Lov Grover, Lucent Technologies
- Grover's Algorithm: Quantum Database Search, C. Lavor, L.R.U. Manssur, R. Portugal
- Grover's algorithm on arxiv.org
- Grover's Algorithm simulation and circuit diagram.
- Grover’s Quantum Search Algorithm animated explanation.
-  simulation and circuit diagram with cats.