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| '''Grover's algorithm''' is a [[quantum algorithm]] for searching an [[sorting|unsorted]] [[database]] with ''N'' entries in ''O''(''N''<sup>1/2</sup>) time and using ''O''(log ''N'') storage space (see [[big O notation]]). [[Lov K. Grover|Lov Grover]] formulated it in 1996.
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| In models of classical computation, searching an unsorted database cannot be done in less than [[linear time]] (so merely [[linear search|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''(''N''<sup>1/2</sup>) is asymptotically the fastest possible for searching an unsorted database in the ''linear'' quantum model.<ref name=bennett_1997>Bennett C.H., Bernstein E., Brassard G., Vazirani U., ''[http://www.cs.berkeley.edu/~vazirani/pubs/bbbv.ps The strengths and weaknesses of quantum computation]''. [[SIAM Journal on Computing]] 26(5): 1510–1523 (1997). Shows the optimality of Grover's algorithm.</ref> 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. Unsorted search speeds of up to [[constant time]] are achievable in the ''nonlinear'' quantum model.<ref>Meyer, D.A. and Wong, T.G. ''[http://arxiv.org/abs/1303.0371 Nonlinear Quantum Search Using the Gross-Pitaevskii Equation].''</ref>
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| 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.)
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| == Applications ==
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| 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''.
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| 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.
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| == Setup ==
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| Consider an unsorted database with ''N'' entries. The algorithm requires an ''N''-dimensional [[mathematical formulation of quantum mechanics|state space]] ''H'', which can be supplied by ''n''=log<sub>2</sub> ''N'' [[qubit]]s. 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]], ''U<sub>ω</sub>'', which acts as:
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| :<math> U_\omega |\omega\rang = - |\omega\rang </math>
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| :<math> U_\omega |x\rang = |x\rang \qquad \mbox{for all}\ x \ne \omega</math>
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| Our goal is to identify the index <math>|\omega\rang</math>.
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| == Algorithm steps==
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| [[File:Grovers algorithm.svg|500px|thumb|right|[[Quantum circuit]] representation of Grover's algorithm]]
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| The steps of Grover's algorithm are given as follows. Let <math>|s\rangle</math> denote the uniform superposition over all states
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| :<math>|s\rang = \frac{1}{\sqrt{N}} \sum_{x=1}^{N} |x\rang</math>.
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| Then the operator
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| :<math>U_s = 2 \left|s\right\rangle \left\langle s\right| - I</math>
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| is known as the Grover diffusion operator.
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| Here is the algorithm:
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| # Initialize the system to the state<br /><math>|s\rang = \frac{1}{\sqrt{N}} \sum_{x=1}^{N} |x\rang </math>
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| # Perform the following "Grover iteration" ''r(N)'' times. The function ''r(N),'' which is asymptotically ''O(N<sup>½</sup>)'', is described below.
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| ## Apply the operator <math>U_\omega</math>.
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| ## Apply the operator <math>U_s</math>.
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| # Perform the measurement Ω. The measurement result will be λ<sub>ω</sub> with probability approaching 1 for N≫1. From λ<sub>ω</sub>, ω may be obtained.
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| == The first iteration ==
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| A preliminary observation, in parallel with our definition
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| :<math> U_s = 2 \left|s\right\rangle \left\langle s\right| - I</math>,
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| is that ''U<sub>ω</sub>'' can be expressed in an alternate way: | |
| :<math> U_\omega = I - 2 \left|\omega\right\rangle \left\langle \omega\right|</math>.
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| To prove this it suffices to check how ''U<sub>ω</sub>'' acts on basis states:
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| :<math> (I-2| \omega\rangle \langle \omega|)|\omega\rang=|\omega\rang-2| \omega\rangle \langle \omega|\omega\rang=-|\omega\rangle = U_\omega |\omega\rang</math>.
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| :<math> (I-2| \omega\rangle \langle \omega|)|x\rang=|x\rang-2| \omega\rangle \langle \omega|x\rang=|x\rangle = U_\omega |x\rang </math> for all <math> x \neq \omega</math>.
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| The following computations show what happens in the first iteration:
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| :<math> \lang\omega|s\rang =\lang s|\omega\rang = \frac{1}{\sqrt{N}} </math>.
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| :<math> \langle s| s\rang =N\frac{1}{\sqrt{N}}\cdot \frac{1}{\sqrt{N}}=1</math>.
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| :<math> U_\omega |s\rang = (I-2| \omega\rangle \langle \omega|)|s\rang=|s\rang-2| \omega\rangle \langle \omega|s\rang=|s\rang-\frac{2}{\sqrt{N}}|\omega\rangle </math>.
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| :<math>U_s(|s\rang-\frac{2}{\sqrt{N}}|\omega\rangle) = (2 |s\rang \lang s| - I)(|s\rang-\frac{2}{\sqrt{N}}|\omega\rangle)=2 |s\rang \lang s|s\rang-|s\rang-\frac{4}{\sqrt{N}}|s\rang \langle s|\omega\rang+\frac{2}{\sqrt{N}}|\omega\rang=</math>
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| :<math>=2|s\rang-|s\rang-\frac{4}{\sqrt{N}}\cdot\frac{1}{\sqrt{N}}|s\rang+\frac{2}{\sqrt{N}}|\omega\rang=|s\rang-\frac{4}{N}|s\rang+\frac{2}{\sqrt{N}}|\omega\rang=\frac{N-4}{N}|s\rang+\frac{2}{\sqrt{N}}|\omega\rang</math>.
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| After application of the two operators ( <math>U_\omega</math> and <math>U_s</math> ), the amplitude of the searched-for element has increased from <math> \left| \lang \omega | s \rang \right|^2 = 1/N</math> to <math> \left| \lang \omega | U_s U_\omega s \rang \right|^2 \approx 9/N</math>.
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| ==Description of ''U<sub>ω</sub>''==
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| Grover's algorithm requires a "quantum oracle" operator <math>U_{\omega}</math> 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 <math>f</math> which returns <math>f(x) = 1</math> if <math>|x\rang</math> is a solution to the search problem and <math>f(x) = 0</math> otherwise. The oracle is a unitary operator which operates on two qubits, the index qubit <math>|x\rang</math> and the oracle qubit <math>|q\rang</math>:
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| :<math>|x\rang|q\rang \overset{U_{\omega}}\longrightarrow |x\rang|q\oplus f(x)\rang</math>
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| As usual, <math>\oplus</math> denotes addition modulo 2. The operation flips the oracle qubit if <math>f(x) = 1</math> and leaves it alone otherwise. In Grover's algorithm we want to flip the sign of the state <math>|x\rang</math> if it labels a solution. This is achieved by setting the oracle qubit in the state <math>(|0\rang - |1\rang)/\sqrt{2}</math>, which is flipped to <math>(|1\rang - |0\rang)/ \sqrt{2}</math> if <math>|x\rang</math> is a solution:
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| :<math>|x\rang\left(|0\rang - |1\rang\right)/\sqrt{2} \overset{U_{\omega}}\longrightarrow (-1)^{f(x)}|x\rang\left( |0\rang- |1\rang\right)/\sqrt{2} </math>
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| We regard <math>|x\rang</math> 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 <math>U_{\omega}</math> is simply written as:
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| :<math>|x\rang \overset{U_{\omega}} \longrightarrow (-1)^{f(x)}|x\rang</math>
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| == Geometric proof of correctness==
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| [[File:Grovers algorithm geometry.png|thumb|310px|Picture showing the geometric interpretation of the first iteration of Grover's algorithm. The state vector <math>|s\rang</math> is rotated towards the target vector <math>|\omega\rang</math> as shown.]]
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| Consider the plane spanned by <math>|s\rang</math> and <math>|\omega\rang</math>; equivalently, the plane spanned by <math>|\omega\rang</math> and the perpendicular ket <math>|s'\rang = \frac{1}{\sqrt{N - 1}}\sum_{x \neq \omega} |x\rang</math>. We will consider the first iteration, acting on the initial ket <math>|s\rang</math>. Since <math>|\omega\rang</math> is one of the basis vectors in <math>|s\rang</math> the overlap is
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| :<math> \lang s'|s\rang = \sqrt{\frac{N-1}{N}} </math>
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| In geometric terms, the angle <math>\theta/2</math> between <math>|s\rang</math> and <math>|s'\rang</math> is given by:
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| :<math> \sin \theta/2 = \frac{1}{\sqrt{N}} </math>
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| The operator <math>U_{\omega}</math> is a reflection at the hyperplane orthogonal to <math>|\omega\rang</math> for vectors in the plane spanned by <math>|s'\rang</math> and <math>|\omega\rang</math>; i.e. it acts as a reflection across <math>|s'\rang</math>. The operator <math>U_s</math> is a reflection through <math>|s\rang</math>. Therefore, the state vector remains in the plane spanned by <math>|s'\rang</math> and <math>|\omega\rang</math> after each application of the operators <math>U_s</math> and <math>U_{\omega}</math>, and it is straightforward to check that the operator <math>U_s U_{\omega}</math> of each Grover iteration step rotates the state vector by an angle of <math>\theta = 2\arcsin \frac{1}{\sqrt{N}} </math>.
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| We need to stop when the state vector passes close to <math>|\omega\rang</math>; after this, subsequent iterations rotate the state vector ''away'' from <math>|\omega\rang</math>, reducing the probability of obtaining the correct answer. The exact probability of measuring the correct answer is:
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| :<math> \sin^2\left( \left( r+ \frac{1}{2} \right)\theta\right)</math>
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| where ''r'' is the (integer) number of Grover iterations.
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| The earliest time that we get a near-optimal measurement is therefore <math>r \approx \pi \sqrt{N} / 4</math>.
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| == Algebraic proof of correctness ==
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| To complete the algebraic analysis we need to find out what happens when we repeatedly apply <math>U_s U_\omega</math>. 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 <math>s</math> and <math>\omega</math>. We can write the action of <math>U_s</math> and <math>U_\omega</math> in the space spanned by <math>\{|s\rang, |\omega\rang\}</math> as:
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| :<math> U_s : a |\omega \rang + b |s \rang \mapsto (|\omega \rang \, | s \rang) \begin{pmatrix}
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| -1 & 0 \\
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| 2/\sqrt{N} & 1 \end{pmatrix}\begin{pmatrix}a\\b\end{pmatrix}.</math>
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| :<math> U_\omega : a |\omega \rang + b |s \rang \mapsto (|\omega \rang \, | s \rang) \begin{pmatrix}
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| -1 & -2/\sqrt{N} \\
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| 0 & 1 \end{pmatrix}\begin{pmatrix}a\\b\end{pmatrix}.</math>
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| So in the basis <math>\{ |\omega\rang, |s\rang \}</math> (which is neither orthogonal nor a basis of the whole space) the action <math>U_sU_\omega</math> of applying <math>U_\omega</math> followed by <math>U_s</math> is given by the matrix
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| :<math> U_sU_\omega = \begin{pmatrix} -1 & 0 \\ 2/\sqrt{N} & 1 \end{pmatrix}
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| \begin{pmatrix}
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| -1 & -2/\sqrt{N} \\
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| 0 & 1 \end{pmatrix}
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| =
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| \begin{pmatrix}
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| 1 & 2/\sqrt{N} \\
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| -2/\sqrt{N} & 1-4/N \end{pmatrix}.</math>
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| This matrix happens to have a very convenient [[Jordan form]]. If we define <math>t = \arcsin(1/\sqrt{N})</math>, it is
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| :<math> U_sU_\omega = M \begin{pmatrix} \exp(2it) & 0 \\ 0 & \exp(-2it)\end{pmatrix} M^{-1}</math> where <math>M = \begin{pmatrix}-i & i \\ \exp(it) & \exp(-it) \end{pmatrix}.</math>
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| It follows that ''r''th power of the matrix (corresponding to ''r'' iterations) is
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| :<math> (U_sU_\omega)^r = M \begin{pmatrix} \exp(2rit) & 0 \\ 0 & \exp(-2rit)\end{pmatrix} M^{-1}</math>
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| Using this form we can use trigonometric identities to compute the probability of observing ''ω'' after ''r'' iterations mentioned in the previous section,
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| :<math>\left|\begin{pmatrix}\lang\omega|\omega\rang & \lang\omega|s\rang\end{pmatrix}(U_sU_\omega)^r \begin{pmatrix}0 \\ 1\end{pmatrix} \right|^2 = \sin^2\left( (2r+1)t\right)</math>.
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| 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 <math>2rt \approx \pi/2</math> or <math>r = \pi/4t = \pi/4\arcsin(1/\sqrt{N}) \approx \pi\sqrt{N}/4</math>. Then the system is in state
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| :<math> (|\omega \rang \, | s \rang) (U_sU_\omega)^r \begin{pmatrix}0\\1\end{pmatrix} \approx (|\omega \rang \, | s \rang) M \begin{pmatrix} i & 0 \\ 0 & -i\end{pmatrix} M^{-1} \begin{pmatrix}0\\1\end{pmatrix} = | w \rang \frac{1}{\cos(t)} - |s \rang \frac{\sin(t)}{\cos(t)}.</math>
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| A short calculation now shows that the observation yields the correct answer ''ω'' with error ''O(1/N)''.
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| == Extension to space with multiple targets ==
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| 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)<sup>1/2</sup>/4'' instead of ''πN<sup>1/2</sup>/4''.
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| There are several ways to handle the case if ''k'' is unknown. For example, one could run Grover's algorithm several times, with
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| :<math> \pi \frac{N^{1/2}}{4}, \pi \frac{(N/2)^{1/2}}{4},
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| \pi \frac{(N/4)^{1/2}}{4}, \ldots </math>
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| 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
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| :<math> \pi \frac{N^{1/2}}{4} \left( 1+ \frac{1}{\sqrt{2}}+\frac{1}{2}+\cdots\right) </math>
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| which is still O(''N<sup>1/2</sup>''). 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 <math>\sqrt{N/k}</math> queries. This fact is used in order to solve the collision problem.<ref name=Boyer>{{Citation| author1=Michel Boyer|author2=Gilles Brassard|author3=Peter Høyer|author4=Alain Tapp|journal=Fortsch. Phys.|volume=46|pages=493–506|year=1998|arxiv=quant-ph/9605034|bibcode = 1998ForPh..46..493B |doi = 10.1002/(SICI)1521-3978(199806)46:4/5<493::AID-PROP493>3.0.CO;2-P }}</ref><ref>{{Citation|
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| author=Andris Ambainis|title=Quantum search algorithms|journal=SIGACT News|volume=35|number=2|pages=22–35|year= 2004|arxiv=quant-ph/0504012|bibcode = 2005quant.ph..4012A }}
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| </ref>
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| ==Quantum partial search==
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| A modification of Grover's algorithm called quantum partial search was described by Grover and Radhakrishnan in 2004.<ref>L.K. Grover and J. Radhakrishnan,''Is partial quantum search of a database any easier?''. [http://arxiv.org/abs/quant-ph/0407122 quant-ph/0407122]</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%, 50-70% or 75-100% percentile.
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| To describe partial search, we consider a database separated into <math>K</math> blocks, each of size <math>b = N/K</math>. 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 <math>N/2</math> to <math>(N-b)/2</math>.
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| Grover's algorithm requires <math>\pi/4\sqrt{N}</math> iterations. Partial search will be faster by a numerical factor which 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>.
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| The global Grover operator acts on the blocks. Essentially, it is given as follows:
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| #Perform <math>j_1</math> standard Grover iterations on the entire database.
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| #Perform <math>j_2</math> local Grover iterations. A local Grover iteration is a direct sum of Grover iterations over each block.
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| #Perform one standard Grover iteration
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| 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|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.
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| ==Optimality ==
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| It is known that Grover's algorithm is optimal. That is, any algorithm that accesses the database only by using the operator U<sub>ω</sub> must apply U<sub>ω</sub> at least as many times as Grover's algorithm.<ref name=bennett_1997 /> This result is important in understanding the limits of quantum computation.
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| If the Grover's search problem was solvable with ''log<sup>c</sup> N'' applications of U<sub>ω</sub>, that would imply that [[NP (complexity class)|NP]] is contained in [[BQP]], by transforming problems in NP into Grover-type search problems. The optimality of
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| Grover's algorithm suggests (but does not prove) that NP is not contained in BQP.
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| The number of iterations for ''k'' matching entries, ''π(N/k)<sup>1/2</sup>/4'', is also optimal.<ref name=Boyer />
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| ==Applicability and Limitations==
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| When applications of Grover's algorithm are considered, it should be emphasized that the database is not represented explicitly. Instead, an oracle is invoked to evaluate an item by its index. Reading a full data-base item by item and converting it into such a representation may take a lot longer than Grover's search. To account for such effects, Grover's algorithm can be viewed as solving an equation or satisfying a constraint. In such applications, the oracle is a way to check the constraint and is not related to the search algorithm. This separation usually prevents algorithmic optimizations, whereas conventional search algorithms often rely on such optimizations and avoid exhaustive search. These and other considerations about using Grover's algorithm are discussed in <ref name=Viamontes>{{Citation| author1=Viamontes G.F.|author2=Markov I.L.|author3=Hayes J.P.|title="Is Quantum Search Practical?"|journal=IEEE/AIP Computing in Science and Engineering|pages=62-70|volume=7|issue=3|year=2005|url=http://web.eecs.umich.edu/~imarkov/pubs/jour/cise05-grov.pdf}}</ref>
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| ==See also==
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| * [[Amplitude amplification]]
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| * [[Shor's algorithm]]
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| ==Notes==
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| <references/>
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| == References ==
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| * Grover L.K.: ''[http://arxiv.org/abs/quant-ph/9605043 A fast quantum mechanical algorithm for database search]'', Proceedings, 28th Annual ACM Symposium on the Theory of Computing, (May 1996) p. 212
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| * Grover L.K.: ''[http://arxiv.org/abs/quant-ph/0109116 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.
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| * Nielsen, M.A. and Chuang, I.L. ''Quantum computation and quantum information''. Cambridge University Press, 2000. Chapter 6.
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| * [http://www.bell-labs.com/user/feature/archives/lkgrover/ What's a Quantum Phone Book?], Lov Grover, Lucent Technologies
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| * [http://arxiv.org/abs/quant-ph/0301079 Grover's Algorithm: Quantum Database Search], C. Lavor, L.R.U. Manssur, R. Portugal
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| * [http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=359266 Grover's algorithm on arxiv.org]
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| ==External links==
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| *[http://www.davyw.com/quantum?example=Grover's%20Algorithm Grover's Algorithm] simulation and circuit diagram.
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| *[http://twistedoakstudios.com/blog/Post2644_grovers-quantum-search-algorithm Grover’s Quantum Search Algorithm] animated explanation.
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| *[http://www.irisa.fr/prive/fschwarz/mit1_algo2_2013/grover_s_algorithm/] simulation and circuit diagram with cats.
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| {{Quantum computing}}
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| {{DEFAULTSORT:Grover's Algorithm}}
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| [[Category:Quantum algorithms]]
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| [[Category:Search algorithms]]
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