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A fast quantum mechanical algorithm for database searchLov K. Grover
3C404A, AT&T Bell Labs600 Mountain AvenueMurray Hill NJ [email protected]
SummaryAnunsorted database contains N records, of which justone satisfies a particular property. The problem is toidentify that one record. Any classical algorithm, deterministic or probabilistic, will clearly take O (N) stepssince on the average it will have to examine a large fraction of the N records. Quantum mechanical systems cando several operations simultaneously due to their wavelike properties. This paper gives an O (JN) step quantum mechanical algorithm for identifying that record. Itis within a constant factor of the fastest possible quantum mechanical algorithm.1. Introduction1.0 Background Quantum mechanical computerswere proposed in the early 1980s [Benioff80] andshown to be at least as powerful as classical computers an important but not surprising result, since classicalcomputers, at the deepest level, ultimately follow thelaws of quantum mechanics. The description of quantum mechanical computers was formalized in the late80s and early 90s [Deutsch85][BB92] [BV93] [Yao93]and they were shown to be more powerful than classicalcomputers on various specialized problems. In early1994, [Shor94] demonstrated that a quantum mechanical computer could efficiently solve a wellknown problem for which there was no known efficient algorithmusing classical computers. This is the problem of integerfactorization, i.e. testing whether or not a given integer,N, is prime, in a time which is a finite power ofo (logN) .  
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This paper applies quantum computing to amundane problem in information processing and presents an algorithm that is significantly faster than anyclassical algorithm can be. The problem is this: there isan unsorted database containing N items out of whichjust one item satisfies a given condition  that one itemhas to be retrieved. Once an item is examined, it is possible to tell whether or not it satisfies the condition inone step. However, there does not exist any sorting onthe database that would aid its selection. The most efficient classical algorithm for this is to examine the itemsin the database one by one. If an item satisfies therequired condition stop; if it does not, keep track of thisitem so that it is not examined again. It is easily seenthat this @gorithm will need to look at an average of\items before finding the desired one.1.1 Search Problems in Computer ScienceEven in theoretical computer science, the typical problem can be looked at as that of examining a number ofdifferent possibilities to see which, if any, of them satisfy a given condition. This is analogous to the searchproblem stated in the summary above, except that usually there exists some structure to the problem, i.e somesorting does exist on the database. Most interestingproblems are concerned with the effect of this structureon the speed of the algorithm. For example the SATproblem of NPcompleteness asks whether it is possibleto find any combination of n binary variables that satisfies a certain set of clauses C, in time polynomial in n. Inthis case there are N=2 possible combinations whichhave to be searched for any that satisfy the specifiedproperty and the question is whether we can do that in atime which is a finite power of O (logN) , i.e. O (nk) .Thus if it were possible to reduce the number of steps toa finite power of O (logIV) (instead of O ( ~N) as inthis paper), it would yield a polynomial time algorithmfor NPcomplete problems.
In view of the fundamental nature of the searchproblem in both theoretical and applied computer science, it is natural to ask  how fast can the basic identifi
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cation problem be solved without assuming anythingabout the structure of the problem? It is generallyassumed that this limit is O (N) since there are N itemsto be examined and a classical algorithm will clearlytake O (N) steps. However, quantum mechanical systems can sometimes be counterintuitive [BV93][Shor94] [Simon94]. This paper presents an O (~~step algorithm for the above problem.
There is a matching lower bound on how fastthe desired item can be identified. [BBBV94] show intheir paper that in order to identify the desired element,without any information about the structure of the database, a quantum mechanical system will need at leastCl ( ~~ steps. Since the number of steps required bythe algorithm of this paper is O ( fi) , it is within aconstant factor of the fastest possible quantum mechanical algorithm.1.2 Quantum mechanical algorithms A goodstarting point to think of quantum mechanical algorithms is probabilistic algorithms [BV93] (e.g. simulated annealing), In these algorithms, instead of havingthe system in a specified state, it is in a disrnbution overvarious states with a certain probability of being in eachstate. At each step, there is a certain probability of making a transition from one state to another. The evolutionof the system is obtained by premultiplying this probability vector (that describes the distribution of probabilities over various states) by a state transition matrix.Knowing the initial distribution and the state transitionmatrix, it is possible in principle to calculate the distribution at any instant in time. ~
Just like classical probabilistic algorithms,quantum mechanical algorithms work with a probabilitydistribution over various states. However, unlike classicat systems, the probability vector does not completelydescribe the system. In order to completely describe thesystem we need the amplitude in each state which is acomplex number. The evolution of the system isobtained by Premultiplying this amplitude vector (thatdescribes the distribution of amplitudes over variousstates) by a transition matrix, the entries of which arecomplex in general. The probabilities in any state aregiven by the square of the absolute values of the amplitude in that state. It can be shown that in order to conserve probabilities, the state transition matrix has to beunitary [BV93].The machinery of quantum mechanical algorithms is illustrated by discussing the three operationsthat are needed in the algorithm of this paper. The tirst isthe creation of a configuration in which the amplitude ofthe system being in any of the 2 basic states of the sys
tem is equal; the second is the Fourier transformationoperation and the third the selective rotation of differentstates.
A basic operation in quantum computing is thatof a fair coin flip performed on a single bit whosestates are O and 1 [Simon93]. This operation is repre
[111sented by the following matrix: M = ~_l .Abit
in the state O is transformed into a superposition in thetwo states: ()$+ . Similarly a bit in the state 1 is
Msfomedkto(;+ioeof iemagnitideoftheamplitude in each state is ~ but the phase of the&litude in the state 1 is inverted. The phase does nothave an analog in classical probabilistic algorithms. Itcomes about in quantum mechanics since the amplitudes are in general complex. As mentioned previously,the probabilities are determined by the square of theabsolute value of the amplitude in each state; hence theprobabilities of the two states in both cases (i.e. when theamplitudes are ()$,$ and when they are
($$) ) are equal. And yet the distributions havevery different properties. For example, applying M tothe disrnbution ()+ i results in the configurationbeing in the state O; applying M to the distribution
(3 d)11, results in the system being in the state 1.Two distributions that had the same probability distributions over all states, now have orthogonal distributions!This illustrates the point that, in quantum mechanics, thecomplete description of the system requires the amplitudes over all states, just the probabilities is not enough.
In a system in which the states are described byn bits (it has 2n possible states) we can perform thetransformation M on each bit independently in sequencethus changing the state of the system. The state transition matrix representing this operation will be of dimension 2n X 2n. In case the initial configuration was theconfiguration with all n bits in the lirst state, the resuRant configuration will have an identical amplitude of
n.22 in each of the 2n states. This is a way of creating adistribution with the same amplitude in all 2 states.
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Next consider the case when the sting stateis another one of the 2 states, i.e. a state described byan n bit binary string with some 0s and some 1s. Theresult of performing the transformation M on each bitwill be a superposition of states described by all possible n bit binary strings with amplitude of each state hav
n2ing a magnitude equal to 2 and sign either + or . Todeduce the sign, observe that from the definition of the[111matrix M, i.e. M = fi l1 , the phase of the resulting configuration is changed when a bit that was previously a 1 remains a 1 after the transformation isperformed. Hence if z be thenbit binary string describing the starting state and ~ the nbit binary stringdescribing the resulting string, the sign of the amplitudeof y is determined by the parity of the bitwise dot product of % and y , i.e. (1) z Y. This transformation isreferred to as the Fourier transformation [DJ92]. Thisoperation is one of the things that makes quantummechanical algorithms more powerful than classicalalgorithms and forms the basis for most significantquantum mechanical algorithms.
The third transformation that we will need isthe selective rotation of the phase of the amplitude incertain states. The transfomnation describing this for a 4
state system is of the form:joooIo o 0 wheree . .
j = = and 4+,$2,03,44 are arbitrary red numbers.Note that, unlike the Fourier transformation and otherstate transition matrices, the probability in each statestays the same since the square of the absolute value ofthe amplitude in each state stays the same.
2. The Abstracted Problem Let a systemhave N = 2n states which are labelted S1 ,S2,...SN These2n states are represented as n bit strings. Let there be aunique state, say Sv, that satisfies the condition C(SV) =1, whereas for all other states S, C(S) = 0 (assume thatfor any state S, the condition C(S) can be evaluated inunit time), The problem is to identify the state SV.
3. Algorithm(i) Initialize the system to the distribution( 1111~~~...~ ) , i.e. there is the same amplitudeto be in each of the N states. It is possible to obtain thisdistribution in O (logN) steps as discussed in section1.2.(ii) Repeat the following unitary operations O (fi)times:
(a) Let the system be in any state S:in case C(S) = 1, rotate the phase byn radians,in case C(S) = O, leave the systemunaltered.
(b) Apply the diffusion transform D whichis defined by the matrix D as follows:Dij = ~ifi#j&Dii=l+~.N NThis diffusion transform, D, can beimplemented as D = FRF, where F theFourier Tkmsform Matrix and R the rotationmatrix are defined as follows:Rij=Oifi#j;Rii= 1 ifi= O; R.. =lifi#O.tlAs discussed in section 1.2, theFourier transformation matrix F isdefined as:Fij = 2n/2 (1) i J, where ~ is thebinary representation of i, andi. j denotes the bitwise dot productof the two n bit strings ~ and ~.
(iii) Sample the resulting state. In case there is aunique state SV such that C(SV) = 1, the final state is SVwith a probability of at least ~.2
4. Outline of rest of paperThe following sections prove that the system discussedin section 3 is indeed a valid quantum mechanical system and that it converges to the desired state with aprobability O(l). The proofs are divided into two parts,First, it is proved that the system is a valid quantummechanical system (theorems 1 & 2) and, second, that itconverges to the desired state (theorems 3, 4 & 5) . Inorder to prove that it is a valid quantum mechanical system we need to prove that it is unitary and that it can beimplemented as a product of local transition matrices.
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The diffusion transform D is defined by the matrix D asfoffow.x(4.0) Dij=; ,ifi#j&Dii = 1+;.Theorem 1 The matix D is unitary.The way D is presented above, it is not a local tmnsitionmatrix since there are transitions from each state to all Nstates. Using the Fourier transformation matrix asdefined in section 3, it can be implemented as a productof three unitary transformations as D = FRF, each ofF & R is a local transition matrix. R as defined in theorem 2 is a phase rotation matrix and is clearly local. Fwhen implemented as in section 1.2 is a local transitionmatrix on each bit.Theorem 2  D can be expressed as D = FRF,where F, the Fourier Transform Matrix and R, the rotation matrix are defined as followsRij=Oifi#j,Rii=lif i= O, Rii=lifi#O.
Next it is proved that the algorithm indeed converges tothe desired state. In order to follow the evolution of thestate when the unitary transformation D of step (ii)(b) ofthe algorithm is applied, we proveTheorem 3 Let the state vector be as follows  forany one state the amplitude is k, for each of the remaining (N1 ) states the amplitude is 1. Then after applyingthe diffusion transform D, the amplitude in the one state
()s kl = ~1 k+ 291 and the amplitude ineach of the remaining (N1) states is11 =$k+~l.
Using this we prove a simple relation between theamplitudes k & 1.Corollary 3.1 Let the state vector be as follows for any one state the amplitude is k, for each of theremaining (N1) states the amplitude is 1.Let k and 1bereal numbers (in general the amplitudes can be complex). Let k be negative and 1 be positive such that~ < ~N. Then after applying the diffusion transformboth k and 1are positive numbers.As is well known, in a unitary transformation the totalprobability is conserved  this is proved for the particu
lar case of the diffusion transformation by using theorem 3.Corollary 3.2 Let the state vector be as follows for the state that satisfies C(S) = 1, the amplitude is k,for each of the remaining (N1) states the amplitude is 1.Then if after applying the diffusion transformation D,the new amplitudes are respectively kl and 11 iis derivedin theorem 3, thenk:+ (N1)1; = k2+ (N1)12
k & 1vary with each iteration of the algorithm in section3. Therefore if we are given k at any step of the algorithm, by using the conservation result of corollary 3.2we can immediately obtain 111, the sign of 1 can bededuced by corollary 3.1 it is thus enough to keep trackof one variable k or 1, not both. Using this we prove Theorem 4 Let the state vector before step (a) of thealgorithm be as follows  for the one state that satisfiesc (s) = 1, the amplitude is k, for each of the remaining (Nl) states the amplitude is 1 such that(o~ . Also after steps (a) and (b), 1>0.2fiUsing this it immediately follows that there exists anumber M less than EN, such that in M repetitions ofthe loop in step (ii), k wifl become ~. Since the proba&bility of the system being found in any particular state isproportional to the square of the amplitude, it followsthat the probability of the system being in tlhe desiredstate when k is ~, 1J2 is k2 = ~. Therefore if the systemis now sampled, it will be in the desired state with aprobability of ~. Finally it is proved that once we candesign an experiment with a finite success probability of(), in O ~ repetitions of the experiment it is possibleto have a probability of success very close to unity.
Theorem 5 Let the probability of a particular outcome in an experiment be &. Then if the experiment berepeated ~ times, the probability that the outcome doesnot happen even once is less than or equal to eK.
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Section 6 quotes the argument fkom [BBBV94] that it isnot possible to identify the desired record in less thanQ (J7i7isteps.
5. ProofsAs mentioned before (4.0), the diffusion transform D isdefined by the matrix D as follows:(5.0) Dij=~, ifi#j&Dii= 1+~.
Theorem 1 The matrix D is unitary.Proof  For a marnx to be unitary it is necessary andsufficient to prove that the columns are orthonormal.The magnitude of each column vector isnED?. j = 1, 2...n. Substituting from (5.0), it folZJyi=l
()ows that this is 1 ~ 2 2+ (N 1):2 which is equalto 1. The dot product of two distinct column vectors isnxDikDij. This is N~ : = O,N2 Ni=lTheorem 2  D can be expressed as D = FRF,where F, the Fourier Transform Matrix and R, the rotation matrix, are defined as followsRij = O ifi#j,Rii=lif i= O, Rii=lifi#O.Fij = 2M (_l) ~~.Proof  We evaluate FRF and show that it is equal toD. As discussed in section 3, Fij = 2/2 (1) i j,where ; is ,the binary representation of i, and i. Jdenotes the bitwise dot product of the two n bit strings ~and ~. R can be written as R = Rl+R2 whereR1 = 1, 1 is the identity matrix and R2 ~ = 2,R2, ij = O if i#O, j#O. By observing thiit MM = Iwhere M is the matrix defined in section 1.2, it is easilyproved that FF=I and hence D ~ = FRIF = I. Wenext evaluate D2 = FR2F. B y standard matrix multiplica
Xon: D2, ad = FabR2, bCFCd. Uskg the definitionbcof R2 and the fact N = 2n, it follows that
all elements of the matrix Dz equaltwo matrices D1 and D2 gives D.
~, the sum of the
Theorem 3 Let the state vector be as follows  forany one state the amplitude is kl, for each of the remaining (N1) states the amplitude is ll. Then after applyingthe diffusion transform D, the amplitude in the one state() N1 ) 11 and the amplitude insk2 = ~ 1 k1+2( Neach of the remaining (N1) states is
Proof Using the definition of the diffusion transform(5.0) (at the beginning of this section), it follows thatk2 = ();1 k1+2~11
()12 = ;2(N2)11ll+~k+NlN1
l%erefortz
Coro]]ary 3.1 Let the state vector be as follows for any one state the amplitude is k, for each of theremaining (N1) states the amplitude is 1.Let k and 1bereal numbers (in general the amplitudes can be complex). Let k be negative and 1be positive and f2, it fol()ows that ; 1 is negative by assumption k is negative and 2(N; 1)1 is positive and hence kl >0.Similarly it follows that since by theorem 3,11 = ~k + ~1, and so if the condition
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~ c ~ is satisfied, then 11>0. If ~ < ~N,
then for N29 the condition ~ < ~ is satisfiedandll>O.
Corollary 3.2 Let the state vector be as follows for the state that satisfies C(S) = 1, the amplitude is k,for each of the remaining (N1) states the amplitude is 1.Then if after applying the diffusion transformation D,the new amplitudes are respectively kl and 11as derivedin theorem 3, thenk:+ (N1)1; =k2+ (N1)12.Proof  Using theorem 3 it follows that
k: = (N 2) 2kZ + ~ (Nj21) 2/2N2_4(N2) (iVl)kl
N2Similarly
4 (N l)2kz(N1)1? = N2
+ 4(N2) (Nl)klN2)2 N 1)12+ N2 .N2(Adding the previous two equations the corollary follows.
Theorem 4 Let the state vector before step (a) of thealgorithm be as follows  for the one state that satisfiesc (s) = 1, the amplitude is k, for each of the remaining (N1) states the amplitude is 1 such that(o~. Also after steps (a) and (b), 1>0.2~Nproof  Denote the initial amplitudes by k and 1, theamplitudes after the phase inversion (step (a)) by kl and11and after the diffusion transform (step (b)) by kz and12.Using theorem 3, it follows that2= (1#)k+2(i)Therefore
()5.1) Ak= k2k=~+2 l~ 1.
hce(o.~.;?~NIn order to prove 12>0, observe that after the phase
inversion (step (a)), kl 0. Furthermore it fol
owsfiomefactso O. Nowraise both sides to the ~ power.Theorem 5 Let the probability of a particular outcome of an experiment be e. Then if the experiment berepeated ~ times, the probability that the outcome doesnot happen even once is less than or equal to eK.Proof  The probability that the outcome does not happen even once in ~ experiments is (1 c) /&, Bylemma 5 this is less than or equal to e.
6. How fast is it possible to find thedesired element? There is a matching lowerIxmnd from the paper [BBBV94] that suggests that it isnot possible to identify the desired element in fewerthan Q ( ~N) steps. This result states that any quantummechanical algorithm running for T steps is cmly sensitive to O (T2) queries (i.e. if there are more possiblequeries, then the answer to at least one can be flippedwithout affecting the behavior of the algorithm). So inorder to correctly decide the answer which is sensitive
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to N queries will take a running time of T = L2 ( ~N) .To see this, assume that C(S) = O for all states and thealgorithm returns the right result. Then, by [BBBV94] ifT< C?( rN) the answer to at least one of the queriesabout C(S) for some S can be flipped without affectingthe result, thus giving an incorrect result this time,
7. Other observations1. As mentioned in the introduction, the algorithmof this paper does not use any Imowledge about theproblem. There exist fast quantum mechanical algorithms that make use of the structure of the problem athand. For example Shors primality testing algorithm[Shor94] for a number N can be viewed as a search ofthe (Nl) numbers smaller than N to see if any of them isa factor of N. This algorithm, by cleverly making use ofsome features of the structure of the problem, is the onlyknown algorithm that solves the problem in O (logIV)steps. It might be possible to combine the search schemeof this paper with [Shor94] and other quantum mechanical algorithms to design even faster algorithms. Alternatively, it might be possible to combine it with efficientdatabase search algorithms that make use of specificproperties of the database.
Associative memories were one of the first applications of synthetic neural networks. There, the motivation was to use as much information as possible aboutthe problem so as to encode and retrieve it most efficiently. It is interesting to speculate about computationalproperties and limits of quantum mechanical neural systems that might achieve both efficient encoding as wellas efficient search. [Penrose89] [Penrose94] mentionthat certain neural sensing systems in the retina andbrain operate in a quantum mechanical framework.Quantum mechanical computation is discussed as onepossible aspect distinguishing a brain from a traditionalcomputer. Noise is a factor which limits quantummechanical computation in the brain and elsewhere.Very recent results [Shor95] have shown the existenceof quantum mechanical error corr~ting cod~ whichenable transmission of data even in the presence ofnoise; design of error free quantum mechanical gatessti ll remains an unsolved problem.2. The algorithm as discussed here assumes a uniquestate that satisfies the desired condition. It can be easilymodified to take care of the case when there are multiplestates satisfying the condition C(S) = 1 and it is requiredto return one of these. Two ways of achieving this are:(i) The first possibility would be to repeat the experiment so that it checks for a range of degeneracy, i.e.redesign the experiment so that it checks for the degen
eracy of the solution being in the range(k, k + 1, . . .2k) for various k. Then within log N repetitions of this procedure, one can ascertain whether ornot there exists at least one state out of the N states thatsatisfies the condition.(ii) The other possibility is to slightly perturb the problem in a random fashion as discussed in [MVV87] sothat with a high probability the degeneracy is removed.There is also a scheme discussed in [VV86] by which itis possible to modify any algorithm that solves an NPsearch problem with a unique solution and use it tosolve an NPsearch problem in general.
T. Acknowledgments The author gratefullyacknowledges Peter Shors and Ethan Bernsteins helpin going through this paper in detail and making severalextremely constructive suggestions.
8. ReferencesBBBV94]
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[Pemose89] R. Penrose, The Emperors New Mind,Concerning Computers, Minds and theLaws of Physics. Oxford UniversityPress, 1989, pp. 400402.
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