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one of the most celebrated applications of quantum information processing is grover s quantum search algorithm , which allows an initially unknown element to be determined from @xmath0 equally likely possibilities in @xmath1 queries @xcite . this outperforms the optimum classical strategy ( a random trial and error of elements ) , which requires @xmath2 steps on average . in addition to providing a speed - up of the unstructured search problem , grover s algorithm can also be adapted to look for solutions to a range of mathematically difficult problems that have structure , by nesting one quantum search inside another @xcite . as with other applications of quantum computing , the benefits of a quantum over a classical approach increase with the size of the database . indeed , it has been suggested that the _ primary resource _ for quantum computation is a hilbert - space dimension , which grows exponentially with the available physical resources @xcite . another important consideration is that the physical and temporal resources required to implement the algorithm do not grow too rapidly with the register size @xmath0 . proof - of - principle quantum search has been successfully demonstrated in nuclear magnetic resonance @xcite , linear optical @xcite and trapped - ion systems @xcite , as well as with individual rydberg atoms @xcite and in classical optics @xcite . of these , only the trapped - ion platform possesses a fully scalable hilbert space and in this sense it is realistically the only candidate for performing a practically useful quantum search . we note , however , that while the trapped - ion system is scalable @xcite , the largest dimensional quantum search so far performed with trapped ions was for a database size of @xmath3 @xcite . extending the approach of ref . @xcite to a large number of ions is highly demanding , since it requires the ability to construct , with very high fidelity , a great number of multiply - conditional gates . for two ions , as in ref . @xcite , each oracle query amounts to a controlled phase gate between the ions . when @xmath4 ions are involved , however , the oracle operator becomes a gate which is multiply - conditional upon the internal state of _ all _ @xmath4 _ ions_. although such a multiply - conditional gate may be decomposed efficiently using a series of one- and two - ion gates @xcite , in practice such a synthesis becomes a daunting task , even for a moderate number of ions . for example , for a four - qubit register , the oracle may be constructed using 13 two - qubit conditional gates @xcite , each of these requiring 5 consecutive physical interactions on average , which is beyond the capabilities of current experiments @xcite . the above example makes it apparent that there is a clear distinction between an approach , which is only _ formally scalable _ , and an approach , which is _ realistically scalable _ using current technology . it is highly desirable , therefore , to find new ways of performing quantum search in a scalable system , which does not require an exorbitant number of elementary physical interactions . in this paper we propose a novel approach to perform grover s search algorithm . the linear crystal of @xmath4 ions is prepared in a symmetric dicke state with @xmath5 excitations , @xmath6 ; hence the register dimension scales _ exponentially _ with the number of ions . the implementation of the quantum search involves a series of red - sideband laser pulses addressing all , or half , of the ions in the trap . thus each of the inversion about average and the oracle query can be produced in a _ single _ interaction step . consequently , the total number of _ physical _ steps is the same as the number of algorithmic steps .    the implementation of the grover s search algorithm proposed below follows earlier proposals , wherein the database scales linearly @xcite ( see also @xcite ) and quadratically @xcite with the number of ions @xmath4 . the database size here , which is @xmath7 , scales exponentially with @xmath4 and it is therefore of fully quantum nature . the reduction of the full hilbert space of dimension @xmath8 by a factor of @xmath9 is compensated by the absence of any ancilla qubits or the need of error correction because of the ultra - compact implementation , in which each logical mathematical step is realized with a single physical interaction ( laser pulse ) . the remainder of this paper is organized as follows . a brief review of grover s algorithm for a quantum mechanical speed - up of the unstructured search problem is given in sec . [ quantumsearch ] . particular attention is given to the ideas of amplitude amplification and generalized reflections , which lie at the heart of the search algorithm . [ system ] introduces a model hamiltonian describing a single laser pulse illuminating a chain of trapped ions , and a convenient hilbert - space factorization , with which to describe the dynamics . by making use of the simplified dynamics in this factorized basis , we construct recipes for the approximate synthesis of the two reflections required for a quantum search , and examples of numerical simulations are presented in sec . [ algorithm ] . in sec . [ conclusions ] we summarize our findings . grover s algorithm provides a method for solving the unstructured search problem , which can be stated as follows : given a collection of database elements @xmath10 , and an _ oracle function _ @xmath11 that acts differently on one _ marked _ element @xmath12 to all others , @xmath13 find the marked element in as few calls to @xmath11 as possible @xcite . if the database is encoded in a physical system that behaves classically , then each oracle query can only act on a single database element . in this case , the optimal search strategy is simply a random selection of elements ; on average , it will be necessary to make approximately @xmath14 calls to the oracle before the marked element @xmath12 is located . the idea underlying grover s algorithm is to encode the database in a physical system that behaves quantum mechanically . therefore , each possible search outcome is represented as a basis vector @xmath15 in an @xmath0-dimensional hilbert space ; correspondingly , the marked element is encoded by a _ marked state _ @xmath16 . hence one can apply unitary operations ( involving the oracle function ) to _ superpositions _ of the different search outcomes . it is thus possible to amplify the amplitude of the marked state @xmath16 using constructive interference , while attenuating all other amplitudes , and locate the marked element in @xmath1 steps . before the execution of the algorithm , the quantum register is prepared in an _ equal _ superposition of all basis elements , @xmath17    central to the operation of the quantum search algorithm is the idea of generalized complex reflections , known in the computer science literature as _ householder reflections _ ( hr ) @xcite : @xmath18 when the phase @xmath19 is set equal to @xmath20 , the effect of @xmath21 on any vector is to invert the sign of the component of this vector along @xmath22 , while leaving all other components unchanged , which amounts to a reflection with respect to an @xmath23-dimensional plane orthogonal to @xmath22 . allowing @xmath19 to take arbitrary values extends the concept of reflection , by imprinting an arbitrary phase onto the component along @xmath22 , rather than a simple sign inversion . householder reflections are widely used in classical data analysis and also constitute a powerful tool for coherent manipulation of quantum systems @xcite . the core component of grover s algorithm is a pair of coupled householder reflections , which together form a single _ grover operator _ @xmath24 : @xmath25 according to standard nomenclature , the operator @xmath26 is referred to as the oracle query , while @xmath27 is known as the inversion - about - average operator . we note that with the initial state given in eq .  , and during successive applications of the operator @xmath24 , the state vector for the system begins and remains in the two - dimensional subspace defined by the non - orthogonal states @xmath16 and @xmath28 . each application of @xmath24 amplifies the marked state population until it reaches a maximum value close to unity after @xmath29 iterations , at which point the search result can be read out by a measurement in the computational basis .    the problem of how to optimize the quantum search routine by allowing arbitrary @xmath30 and @xmath31 has been studied extensively @xcite . it is found that the maximum possible amplitude amplification per step of the marked state arises when the phases @xmath30 and @xmath31 are both set to @xmath20 ( as in grover s original proposal @xcite ) . the corresponding minimum number of search steps @xmath32 is given by : @xmath33   \overset{\mathcal{n}\gg 1}{\sim } \left[\frac{\pi\sqrt{\mathcal{n}}}{4}\right],\ ] ] where @xmath34 $ ] denotes the integer part of @xmath35 . however , this choice of phases is not unique . for large @xmath0 , as long as the _ phase matching _ condition @xmath36 is satisfied @xcite , a high fidelity search can be performed for any value of @xmath37 in the range @xmath38 and for certain values of @xmath37 , a deterministic quantum search is possible @xcite .    in the following two sections , we shall describe how to perform a physically efficient quantum search using the dynamic symmetries in a system of trapped ions . we consider @xmath4 ions confined and laser cooled in a linear paul trap , each with two relevant internal states @xmath39 and @xmath40 , with respective transition frequency @xmath41 . the linear ion crystal interacts with a laser pulse with frequency @xmath42 , where @xmath43 is the axial trap frequency and @xmath44 is the laser detuning from the first red - sideband resonance . we assume that the phonon spectrum can be resolved sufficiently well that only the center - of - mass mode is excited by this interaction and that other vibrational modes can safely be neglected . in the lamb - dicke limit , and after making the optical and vibrational rotating - wave approximations , the interaction hamiltonian is @xcite : @xmath45 . \label{hamiltonian}\ ] ] here @xmath46 and @xmath47 are the raising and lowering operators for the internal states of the @xmath48th ion , @xmath49 is the pauli spin matrix and @xmath50 and @xmath51 are respectively the creation and annihilation operators of center - of - mass phonons . the coupling between the internal and motional degrees of freedom is @xmath52 , where @xmath53 is the single - ion lamb - dicke parameter , with @xmath54 being the laser wave vector and @xmath55 is the mass of the ion . the function @xmath56 is the real - valued time - dependent rabi frequency . the hamiltonian conserves the total number of excitations ( @xmath57 ionic plus @xmath58 motional ) , which in the scheme we propose , is half the number of ions , i.e. @xmath5 ( with @xmath4 even ) , @xmath59 . the energy pattern splits into manifolds corresponding to @xmath60 ionic and @xmath61 motional excitations . each manifold is @xmath62-fold degenerate , where @xmath63 . it is readily verified that the dimension of the @xmath64 manifold , @xmath65 , grows _ exponentially _ with @xmath4 ; indeed , for large @xmath4 we have [ database size ] c_n/2^n~ . the subspace of the overall hilbert space , which spans the manifold @xmath65 , we shall use to represent the state of the register in sec . [ algorithm ] ; @xmath65 is the set of states , which encode the database of @xmath0 elements .       ions with 3 excitations consists of a series of independent chainwise linkages . since the total number of excitations is half the number of ions , then from eq . follows that states with @xmath66 are inaccessible and are therefore not shown . the states that make up the longest ladder are all symmetric dicke states @xcite . the number of motional @xmath67 and ionic @xmath60 excitations for each level can be inferred from @xmath68 , eq . . ]    it is possible to move to a new basis in which the hilbert space is factorized into a collection of noninteracting chains of states . the new states we shall call ms states , since they can be obtained by the multilevel morris - shore ( ms ) factorization @xcite .    to determine the ms states first we rewrite the hamiltonian in terms of the total ionic pseudospin : @xmath69 where @xmath70 and @xmath71 . the ms basis consists of the set of the eigenvectors of the two commuting operators @xmath72 and @xmath73 , where @xmath74 . therefore each ms state is assigned two quantum numbers , respectively @xmath75 and @xmath68 . since @xmath72 commutes also with the hamiltonian , the hilbert space factorizes into a set of decoupled chains with different values of @xmath75 . the ms states are sketched in fig . [ fig1 ] for 6 ions with 3 excitations . because the laser pulse couples equally to every ion in the trap , the longest chain is comprised of the symmetric dicke states @xmath76 , wherein a given number of ionic excitations @xmath57 is shared evenly amongst all the ions in the trap @xcite . each chain is assigned different @xmath75 and is comprised of states with different @xmath68 , which varies from 0 to @xmath75 , corresponding to the number of excited ions . if @xmath60 of the ions are excited , then @xmath77 since the total number of excitations is half the number of ions , then from eq . it follows that for each chain @xmath68 has a maximum value of 0 , i.e. states with @xmath66 are inaccessible . by analogy with the traditional angular momentum operators , it follows that the number of states in a chain is equal to @xmath78 ( @xmath75 of the states are not accessible ) , from which @xmath75 can be inferred . the longest chain corresponds to @xmath79 , the next longest to @xmath80 and so on , and overall , there are @xmath81 $ ] chains of different length in the factorized coupling scheme .    for the following analysis , it will be necessary to go further and calculate the couplings in the ms basis . the coupling between the neighbors @xmath82 and @xmath83 follows immediately from the matrix elements of the operators @xmath50 and @xmath84 , i.e. @xmath85    as illustrated in fig . [ fig1 ] , there can be many different degenerate ms states with the same values of @xmath86 and @xmath87 . we label the states with a given @xmath88 and @xmath89 with @xmath90 , where @xmath91 , with @xmath92 . the manifold @xmath65 , which encodes the database elements , consists of the states , for which the pseudo - angular momentum projection is @xmath93 , eq . . this requires that the total number of ions @xmath4 is even . half of these are in state @xmath40 , while the other half are in state @xmath39 ( @xmath94 ) . the number of elements @xmath0 in the database therefore scales exponentially with the number of ions @xmath4 , eq .  . the elements @xmath15 in eq .  , which belong to @xmath65 , can now be written as @xmath95 where the subscript @xmath96 runs over all distinct permutations @xmath97 of the ions internal states . the initial state @xmath28 is thus a symmetric dicke state @xmath6 of @xmath4 ions sharing @xmath5 excitations , @xmath98 our proposed experimental procedure consists of four simple operations . ( i ) the ions are firstly initialized in the entangled dicke state . this may be achieved using very simple adiabatic passage techniques , involving either a pair of chirped laser pulses @xcite or two pairs of delayed but overlapping laser pulses @xcite , and using global addressing . ( ii ) synthesis of the inversion - about - average operator is appealingly simple : @xmath99 is a single red - sideband off - resonant laser pulse addressing all ions in the linear chain . ( iii ) the oracle query @xmath100 is also a single red - sideband laser pulse , applied on half of the ions . after an appropriate number of iterations , the system evolves into the marked state @xmath16 , which can be identified by performing ( iv ) a fluorescence measurement on the entire chain .      in most existing proposals for implementing grover s search algorithm using trapped ions the generation of @xmath99 requires a large number of concatenated physical interaction steps , even for moderate register size @xmath0 . however , by restricting the dynamics to the subspace of the overall hilbert space in which only half of the ions are excited , this operator becomes possible to synthesize in only a _ single _ interaction step . this simplification is achieved by taking advantage of the fact that both the hamiltonian and the state @xmath6 are symmetric under exchange of any two ions . the energies of the ms states do not cross in time so that in the limit @xmath101 the transitions between the ms states vanish due to the effect of adiabatic complete population return @xcite . each of the ms states acquires a phase shift @xmath102 ( the index @xmath75 corresponds to the eigenvalue of @xmath72 ) and the unitary propagator within the dicke manifold @xmath65 is a product of @xmath103 coupled reflections @xcite @xmath104 with @xmath105 and @xmath106 . for a given value of the coupling , the detuning may be adjusted in order to control the phases @xmath102 . ideally , these interaction parameters would be chosen such that @xmath107 ( e.g. , @xmath108 ) , while @xmath109 as this would result in @xmath110 being identical to the inversion - about - average operator @xmath111 . the effect of each oracle query @xmath112 is to imprint a phase shift @xmath113 on the marked state @xmath16 , whilst leaving all other computational basis states unchanged . in general this can be achieved by a multiply - conditional phase gate upon the internal state of all @xmath4 ions in the trap . when more than a few ions are involved , this becomes a prohibitively complicated operation , which generally requires an immense number of one- and two - qubit gates @xcite . however , since we do not work in the whole hilbert space , but rather in the manifold @xmath65 , the oracle operator can be implemented in a much simpler fashion  by a _ single _ red - sideband laser pulse , addressing uniformly those @xmath5 ions in the trap , which share the excitation of the marked state . let s consider an example , when the marked state is @xmath114 . then we address the first three ions . since the initial state @xmath6 and the interactions with the laser are symmetric under exchange of the first three ions , and the last three , as well , during the execution of the algorithm the state of the system is a linear combination of the states @xmath115 , @xmath116 ; the marked state is @xmath117 . because the hamiltonian , which describes the oracle call , does not change the total number of excitations of the first ( and the last ) three ions , i.e. it does not drive the system between @xmath118 and @xmath119 for @xmath120 , in the adiabatic limit the states @xmath121 acquire only phase shifts @xmath122 . therefore the propagator in the manifold @xmath65 is formally described by the action of @xmath5 coupled reflections , @xmath123 hence , as in the case of the inversion - about - average operator , we need to control @xmath5 phases . the interaction parameters should be chosen such that @xmath124 , while @xmath125 , as this would result in @xmath126 being identical to the oracle operator @xmath112 . the phase conditions for the propagators @xmath127 and @xmath128 , derived above , can not be satisfied exactly , since the phases @xmath129 and @xmath122 are overly commensurate . however one can still perform the algorithm with sufficiently high fidelity @xmath130 here @xmath131 is the state of the system after the completion of the algorithm and @xmath132 is its deviation due to the phases deviation . fortunately , a fourier expansion reveals that phase deviation of all odd orders leads to the purely imaginary bracket @xmath133 and hence do not contribute to eq .  . as a result , the algorithm is less sensitive to phase deviations since only those of second order have a leading affect on the final state populations . ions and @xmath5 excitations ( register dimension @xmath134 ) , for @xmath135 ( upper frame ) , @xmath136 ( middle frame ) and @xmath137 ( lower frame ) . the system of ions is assumed to be initialized in the symmetric dicke state @xmath138 . the laser pulses have a gaussian shape , @xmath139 . the thin vertical lines display the timing of the oracle call . the computed numerically scaled detunings and peak rabi frequencies for the oracle and the inversion - about - average operator , respectively , are given in the table below . the marked - state population ( the fidelity ) of around @xmath140 is obtained after @xmath141 , 6 and 12 steps , respectively , in exact agreement with grover s value . the numerical simulation includes all off - resonant transitions to states with @xmath142 . ] [ cols=">,>,>,^,^,^,^ " , ]     [ fig2 ]      as a demonstration , we have solved the schrdinger equation numerically for a gaussian pulse shape and a constant detuning @xmath44 . sample results are shown in fig . [ fig2 ] for @xmath135 , 8 and 10 ions with 3 , 4 and 5 excitations , respectively , which imply databases of @xmath143 , 70 and 252 elements . the fidelity plotted on the vertical axis is the time - dependent population of the marked state . the system of ions is assumed to be prepared initially in the dicke superposition @xmath6 of ionic collective states , each of which contains exactly @xmath5 ion qubits in state @xmath40 and @xmath5 in state @xmath39 . each grover iteration consists of a phase shift for the marked state ( oracle call ) , which amounts a single red - sideband laser pulse , addressing uniformly those @xmath5 ions in the chain , which share the excitation of the marked state , followed by a single red - sideband pulse , which addresses uniformly the entire ion chain ( the inversion - about - average operator ) . the number of steps , for which the algorithm singles out the marked item with a probability of around @xmath140 , is respectively @xmath141 , 6 and 12 as predicted by eq .  . despite the intense flurry of theoretical and experimental activity in the decade following grover s original proposal for a quantum speed - up of unstructured search , a large discrepancy still remains between the current experimental state - of - the - art and what is required for a practically useful quantum search . to highlight this incongruence , we note that the only physical system in which a scalable quantum search has been performed is in a chain of trapped ions , and in that experiment , the search - space totalled just four elements . an important intermediate goal on the long road to performing a practically useful quantum search is to demonstrate grover s algorithm in a moderately - sized trapped - ion quantum register . the standard approach of building up the many - ion grover operator using a network of single- and two - qubit gates is inappropriate for this task , since the required physical resources far exceed those available in today s experiments . by contrast , in this article , we have proposed to construct the many - ion grover operator using only two single off - resonant laser pulses , with a suitably chosen peak rabi frequencies and detunings , which synthesize the inversion - about - average and oracle operators , each in a single shot . a simple recipe for synthesizing the inversion - about - average operator was derived by factorizing the overall hilbert space into a series of independent ladders . the coupling strengths between the ms states were determined solely through a consideration of the angular - momentum structure of the combined ionic pseudospin . the technique proposed in this paper raises the prospect of demonstrating grover s algorithm in a moderately sized trapped - ion database comprising up to several hundred elements , and which scales exponentially with the number of ions ; this is a necessary step on the path to demonstrating a practically useful quantum search , which remains a long - term goal . this work is supported by european commission s projects emali and fastquast , the bulgarian science fund grants vu - f-205/06 , d002 - 90/08 , sofia university grant 020/2009 , and elite programme of the landesstiftung - baden - wrttemberg .
we propose a scalable implementation of grover s quantum search algorithm in a trapped - ion quantum information processor . 
 the system is initialized in an entangled dicke state by using simple adiabatic techniques . 
 the inversion - about - average and the oracle operators take the form of _ single _ off - resonant laser pulses , addressing , respectively , all and half of the ions in the trap . 
 this is made possible by utilizing the physical symmetries of the trapped - ion linear crystal . 
 the physical realization of the algorithm represents a dramatic simplification : each logical iteration ( oracle and inversion about average ) requires only _ two _ physical interaction steps , in contrast to the large number of concatenated gates required by previous approaches . 
 this does not only facilitate the implementation , but also increases the overall fidelity of the algorithm .