Grover's algorithm
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Grover's algorithm is a quantum algorithm for searching an unsorted database with N entries in O(N^{1/2}) time and using O(logN) storage space (see big O notation). It was invented by Lov Grover in 1996.
Contents 
Introduction
Classically, searching an unsorted database requires a linear search, which is O(N) in time. Grover's algorithm, which takes O(N^{1/2}) time, is the fastest possible quantum algorithm for searching an unsorted database. It provides "only" a quadratic speedup, unlike other quantum algorithms, which can provide exponential speedup over their classical counterparts. However, even quadratic speedup is considerable when N is large.
Like all quantum computer 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.
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, Grover's 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. In addition, it can be used to solve NPcomplete problems by performing exhaustive searches over the set of possible solutions. This would result in a considerable speedup over classical solutions, even though it does not provide the "holy grail" of a polynomialtime solution.
Below, we present the basic form of Grover's algorithm, which searches for a single matching entry. The algorithm can be further optimized if there is more than one matching entry and the number of matches is known beforehand.
Setup
Consider an unsorted database with N entries. The algorithm requires an Ndimensional state space H, which can be supplied by log_{2}N qubits.
Let us number the database entries by 0, 1, ... (N1). Choose an observable, Ω, acting on H, with N distinct eigenvalues whose values are all known. Each of the eigenstates of Ω encode one of the entries in the database, in a manner that we will describe. Denote the eigenstates (using braket notation) as
 <math>\{0\rang, 1\rang, \cdots, N1\rang\}<math>
and the corresponding eigenvalues by
 <math>\{\lambda_0, \lambda_1, \cdots, \lambda_{N1} \}<math>
We are provided with a unitary operator, U_{ω}, which acts as a subroutine that compares database entries according to some search criterion. The algorithm does not specify how this subroutine works, but it must be a quantum subroutine that works with superpositions of states. Furthermore, it must act specially on one of the eigenstates, ω>, which corresponds to the database entry matching the search criterion. To be precise, we require U_{ω} to have the following effects:
 <math> U_\omega \omega\rang =  \omega\rang <math>
 <math> U_\omega x\rang = x\rang \qquad \mbox{for all}\ x \ne \omega<math>
Our goal is to identify this eigenstate ω>, or equivalently the eigenvalue ω, that U_{ω} acts specially upon.
Steps of the algorithm
The steps of Grover's algorithm are as follows:
 Initialize the system to the state
 <math>s\rang = \frac{1}{\sqrt{N}} \sum_x x\rang <math>
 Perform the following "Grover iteration" r(N) times. The function r(N) is described below.
 Apply the operator <math>U_\omega<math>
 Apply the operator <math>U_s = 2 \lefts\right\rangle \left\langle s\right  I<math>.
 Perform the measurement Ω. The measurement result will be λ_{ω} with probability approaching 1 for N>>1. From λ_{ω}, ω may be obtained.
Explanation of the algorithm
Our initial state is
 <math> s\rang = \frac{1}{\sqrt{N}} \sum_x x\rang <math>
Consider the plane spanned by s> and ω>. Let ω^{×}> be a ket in this plane perpendicular to ω>. Since ω> is one of the basis vectors, the overlap is
 <math> \lang\omegas\rang = \frac{1}{\sqrt{N}} <math>
In geometric terms, there is an angle (π/2  θ) between ω> and s>, where θ is given by:
 <math> \cos (\frac{\pi}{2}  \theta) = \frac{1}{\sqrt{N}} <math>
 <math> \sin \theta = \frac{1}{\sqrt{N}} <math>
The operator U_{ω} is a reflection at the hyperplane orthogonal to ω>; for vectors in the plane spanned by s> and ω>, it acts as a reflection at the line through ω^{×}>. The operator U_{s} is a reflection at the line through s>. Therefore, the state vector remains in the plane spanned by s> and ω> after each application of U_{s} and after each application of U_{ω}, and it is straightforward to check that the operator U_{s}U_{ω} of each Grover iteration step rotates the state vector by an angle of 2θ toward ω>.
We need to stop when the state vector passes close to ω>; after this, subsequent iterations rotate the state vector away from ω>, reducing the probability of obtaining the correct answer. The number of times to iterate is given by r. In order to align the state vector exactly with ω>, we need:
 <math>\frac{\pi}{2}  \theta = 2 \theta r <math>
 <math>r = \frac{(\frac{\pi}{\theta}  2)}{4} <math>
However, r must be an integer, so generally we can only set r to be the integer closest to (π/θ  2)/4. The angle between ω> and the final state vector is O(θ), so the probability of obtaining the wrong answer is O(1  cos^{2}θ) = O(sin^{2}θ).
For N>>1, θ ≈ N^{1/2}, so
 <math>r \rightarrow \frac{\pi \sqrt{N}}{4} <math>
Furthermore, the probability of obtaining the wrong answer becomes O(1/N), which goes to zero for large N.
Extensions
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 πN^{1/2}/4. There are several ways to handle the case if k is unknown. For example, one could run Grover's algorithm several times, with
 <math> \pi \frac{N^{1/2}}{4}, \pi \frac{(N/2)^{1/2}}{4},
\pi \frac{(N/4)^{1/2}}{4}, \ldots <math>
iterations. For any k, one of iterations will find a matching entry with a sufficiently high probability. The total number of iterations is at most
 <math> \pi \frac{N^{1/2}}{4} \left( 1+ \frac{1}{\sqrt{2}}+\frac{1}{2}+\cdots\right) <math>
which is still O(N^{1/2}).
It is known that Grover's algorithm is optimal. That is, any algorithm that accesses the database only by using the operator U_{ω} must apply U_{ω} at least as many times as Grover's algorithm (Bernstein et al., 1997).
References
 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 [(available online) (http://arxiv.org/abs/quantph/9605043A)]
 Grover L.K.: From Schrodinger's equation to quantum search algorithm, American Journal of Physics, 69(7): 769777, 2001. Pedagogical review of the algorithm and its history.
 http://www.belllabs.com/user/feature/archives/lkgrover/
 Bennett C.H., Bernstein E., Brassard G., Vazirani U., The strengths and weaknesses of quantum computation. SIAM Journal on Computing 26(5): 15101523 (1997). Shows the optimality of Grover's algorithm.