Quantum computers can search rapidly by using almost any transformation

Transcription

1 Quatum computers ca search rapidly by usig almost ay trasformatio Lov K. Grover, 3C-404A Bell Labs, 600 Moutai Aveue, Murray Hill NJ Summary A quatum computer has a clear advatage over a classical computer for exhaustive search. The quatum mechaical algorithm for exhaustive search was origially derived by usig subtle properties of a particular quatum mechaical operatio called the Walsh-Hadamard (W-H) trasform. This paper shows that this algorithm ca be implemeted by replacig the W-H trasform by almost ay quatum mechaical operatio. This leads to several ew applicatios where it improves the umber of steps by a square-root. It also broades the scope for implemetatio sice it demostrates quatum mechaical algorithms that ca readily adapt to available techology. 0. Itroductio Quatum mechaical systems ca be i a superpositio of computatioal states ad hece simultaeously carry out multiple computatios i the same computer. I the last few years there has bee extesive research o how to use this quatum parallelism to carry out meaigful computatios. I ay quatum mechaical computatio the system is iitialized to a state that is easy to prepare ad caused to evolve uitarily, the aswer to the computatioal problem is deduced by a fial measuremet that projects the system oto a uique state. The amplitude (ad hece probability) of reachig a specified fial state depeds o the iterferece of all paths that take it from the iitial to the fial state - the system is thus very sesitive to ay magitude or phase disturbaces that affect ay of the paths leadig to the desired fial state. As a result quatum mechaical algorithms are very delicate ad it is believed a actual implemetatio would eed elaborate procedures for correctig errors [Err]. This paper shows that the quatum search algorithm is surprisigly robust to certai kids of perturbatios. It was origially derived by usig the W-H trasform ad appeared to be a cosequece of the special properties of this trasform, this paper shows that similar results are obtaied by substitutig ay uitary trasformatio i place of the W-H trasform. Sice all quatum mechaical operatios are uitary, this meas that ay quatum mechaical system ca be used - all that is eeded is a valid quatum mechaical operatio ad a way of selectively ivertig the phase of states. Meaigful computatio ca hece be carried out o the basis of uiversal properties of quatum mechaical operatios, this is somewhat similar i spirit to [Neural] where circuit behavior of a certai class of eural etworks was idepedet of the precise ature of the oliearity i each euro.. Quatum operatios I a quatum computer, the logic circuitry ad time steps are essetially classical, oly the memory bits that hold the variables are i quatum superpositios - these are called qubits. There is a set of distiguished computatioal states i which all the bits are defiite 0s or s. I a quatum mechaical algorithm, the quatum computer cosistig of a umber of qubits, is prepared i some simple iitial state, ad caused to evolve uitarily for some time, ad the is measured. The algorithm is the desig of the uitary evolutio of the system. Opera-

2 tios that ca be carried out i a cotrolled way are uitary operatios that act o a small umber of qubits i each step. Two elemetary uitary operatios preseted i this sectio are the W-H trasformatio ad the selective iversio of the amplitudes of certai states. A basic operatio i quatum computig is the operatio M performed o a sigle qubit - this is represeted by the followig matrix: M the state 0 is trasformed ito a superpositio: , Similarly state is trasformed ito the superpositio , I a system i which the states are described by qubits (it has N = possible states) we ca perform the trasformatio M o each qubit idepedetly i sequece thus chagig the state of the system. The state trasitio matrix represetig this operatio will be of dimesio. Cosider a case whe the startig state is oe of the basis states, i.e. a state described by a geeral strig of biary digits composed of some 0s ad some s. The result of performig the trasformatio M o each qubit will be a -- superpositio of states cosistig of all possible bit biary strigs with amplitude of each state beig ±. This trasformatio is referred to as the W-H trasformatio [DJ]. This operatio (or a closely related operatio called the Fourier Trasformatio [Factor]) is oe of the thigs that makes quatum mechaical algorithms more powerful tha classical algorithms ad forms the basis for most sigificat quatum mechaical algorithms. The other trasformatio that we will eed is the selective iversio of the phase of the amplitudes i certai states. Ulike the W-H trasformatio ad other state trasitio matrices, the probability i each state stays the same sice the square of the absolute value of the amplitude i each state stays the same. Its realizatio is particularly straightforward; based o [BBHT], we give such a realizatio. Assume that there is a biary fuctio f ( x) that is either 0 or. Give a superpositio over states x, it is possible to desig a quatum circuit that will selectively ivert the amplitudes i all states where f ( x) =. This is achieved by appedig a acilla bit, b ad cosiderig the quatum circuit that trasforms a state x, b ito x, f ( x)xor b (such a circuit exists sice, as proved i [Reversible], it is possible to desig a quatum mechaical circuit to evaluate ay fuctio f ( x) that ca be evaluated classically). If the bit b is iitially placed i a superpositio ( 0 ), this circuit will ivert the amplitudes precisely i the states for which f ( x) =, while leavig amplitudes i other states uchaged.. Amplitude amplificatio A fuctio f ( x), x =, N, is give which is kow to be o-zero at a sigle value of x, say at x = τ - the goal is to fid τ. If there was o other iformatio about f ( x) ad oe were

3 N usig a classical computer, it is easy to see that o the average it would take --- fuctio evaluatios to solve this problem successfully. However, quatum mechaical systems ca explore multiple states simultaeously ad there is o clear lower boud o how fast this could be doe. [BBBV] showed by usig subtle argumets about uitary trasforms that it could ot be doe i fewer tha Ω( N ) steps - subsequetly [Search] foud a algorithm that took exactly O( N ) steps. The basic idea of [Search] was to cosider a N state quatum mechaical system ad map each value of x to a basis state of the system. The system was iitialized so that there was a equal amplitude i each basis state, the by a series of uitary operatios, the amplitude i the state correspodig to x = τ is icreased (this state is deoted as the τ state). A measuremet is the made due to which the system collapses to a basis state, the observed basis state the gives the solutio to the problem with a high probability. This algorithm was based o subtle properties of the W- H trasform. The aalysis i this sectio shows that very similar results are obtaied by replacig the W-H trasform by ay arbitrary uitary operatio. Some of the cosequeces of this are preseted i the ext sectio. Assume that we have at our disposal a uitary operatio U ad we start with the system i a basis state that is easy to prepare, say γ. If we apply U to γ, the amplitude of reachig state τ is U τγ, if we were to observe the system at this poit, the probability of gettig the right state would be U τγ - accordig to the otatio, γ deotes the iitial state ad τ the target state. It would therefore take Ω repetitios of this experimet before a sigle success. This sectio shows how it is possible to reach state τ i oly O steps. This leads to a sizable U τγ improvemet i the umber of steps if U τγ «. Deote the uitary operatio that iverts the amplitude i a sigle basis state x by I x. I matrix otatio this is the diagoal matrix with all diagoal terms equal to, except the xx term which is - a quatum mechaical implemetatio of this was preseted at the ed of sectio. Cosider the followig uitary operator: Q I γ U - ote that sice U is uitary, U is equal to the adjoit (the complex cojugate of the traspose) of U. We first show that Q preserves the two dimesioal vector space spaed by: γ ad U τ (ote that i the situatio of iterest, whe U τγ is small, these two vectors are almost orthogoal).. O( f ( x) ) meas asymptotically less tha a costat times f ( x), Ω( f ( x) ) meas asymptotically greater tha a costat times f ( x). 3

4 First cosider Q γ. By the defiitio of Q, this is: I γ U γ. Note that x x, where x is a basis state, is a N N square matrix all of whose terms are zero, except the xx term which is. Therefore I τ I τ τ & I γ I γ, it follows: () Q γ = ( I γ )U ( I τ τ )U γ = ( I γ ) U U γ + ( I γ )U τ τ U γ Usig the facts: U U = I ad γ, it follows that: () Q γ = γ + ( I γ )U ( τ τ )U γ. Simplifyig the secod term of () by the followig idetities: τ U γ U τγ & U * τ U τγ (as metioed previously, U is uitary ad so U is equal to the complex cojugate of its traspose): (3) Q γ = γ ( 4 U τγ ) + U τγ ( U τ ) Next cosider the actio of the operator Q o the vector U τ. Usig the defiitio of Q (i.e. Q I γ U ) ad carryig out the algebra as i the computatio of Q γ above, this yields: (4) Q( U τ ) I γ U( U τ ) = I γ τ = I γ U τ. Writig I γ as I γ I γ ad as i (3), U * τ U τγ : (5) Q( U τ ) U τ + γ ( U τ ) U * = = τ U τγ γ. It follows that the operator Q trasforms ay superpositio of the vectors γ & U τ ito aother superpositio of the same two vectors, thus preservig the two dimesioal vector space spaed by γ & U τ. (3) & (5) may be writte as: (6) Q γ U τ = ( 4 U τγ ) U τγ * U τγ γ U τ π π This yields a process with a periodicity of as i [BBHT]. If we start with γ, the after repetitios of U τγ 4 U τγ Q we get the superpositio defied by U τ. From this, with a sigle applicatio of U, we ca get τ. Therefore i O steps, we ca start with γ ad reach the target state τ with certaity. The above derivatio easily exteds to the case whe the amplitudes i states, γ & τ, istead of beig iverted 4

5 by I γ & I τ, are rotated by arbitrary phases. However, the umber of operatios required to reach τ will be greater. Give a choice, it would be clearly better to use the iversio rather tha a differet phase rotatio. Also the aalysis ca be exteded to iclude the case where I τ is replaced by V I τ V, V is a arbitrary uitary matrix. The aalysis is the same as before but istead of the operatio U, we will ow have the operatio VU. 3. Examples of quatum mechaical algorithms The iterestig feature of this paper is that ca be ay uitary trasformatio. Clearly, it ca be used to desig algorithms where U is a trasformatio i a quatum computer - this paper give a few such applicatios. The N = states to be searched are represeted by qubits. Accordig to the framework of sectio, a search of the N states ca be carried out quatum mechaically, if we have a uitary operatio U which has a fiite amplitude U τγ to go from the startig state γ to the target state τ. Such a search will take O steps. The followig aalyses calculate U ad thus the umber of steps required U τγ τγ for various algorithms - (i) & (ii) use the W-H trasform as U ; (iii) uses a differet uitary operatio. (i) Exhaustive search startig from the 0 state I case the startig state γ be the 0 state ad the uitary trasformatio U is chose to be W (the W-H trasformatio as discussed i sectio ), the U τγ for ay target state t is Sectio gives a algorithm requirig O , i.e. O( N ) steps. This algorithm would carry out N repeated operatios of Q, with U = U = W, Q becomes Q I W I ; the operatio sequece is hece: 0 τ W ( I W I By rearragig paretheses ad shiftig mius sigs, this may be see to 0 τ W )( I W I 0 τ W )( I W I 0 τ W ) cosist of alteratig repetitios of W I W & I. 0 τ The operatio sequece W I W is simply the iversio about average operatio [Search]. To see this, write I 0 0 U as I 0 0. Therefore for ay superpositio x : W I W x = W ( I 0 0 )W x = x + W 0 0 W x. It 0 is easily see that W 0 0 W x is aother vector each of whose compoets is the same ad equal to A where N A --- N x i i = (the average value of all compoets). Therefore the i th compoet of W I W x is simply: 0 ( x i + A). This may be writte as A + ( A x i ), i.e. each compoet is as much above (below) the average as it was iitially below (above) the average, which is the iversio about average [Search]. 5

6 (ii) Exhaustive search startig from a arbitrary state For the W-H trasform, betwee ay pair of states γ & τ is ± We ca start with ay state γ ad the aalysis of sectio, gives us a algorithm to reach τ N i O , i.e. O( N ), iteratios. Therefore istead of startig with the 0 state, as i (i), we could equally well start with ay state γ, ad repeatedly apply the operatio sequece Q = I γ W I τ W to obtai a equally efficiet U τγ O( N ) algorithm. The dyamics is similar to (i); however, there is o loger the coveiet iversio about average iterpretatio. (iii) Search whe a item ear the desired state is kow Problem: Assume that a bit word is specified - the desired word differs from this i exactly k bits. Solutio The effect of this costrait is to reduce the size of the solutio space. Oe way of makig use of this costrait, would be to map this to aother problem which exhaustively searched the reduced space usig (i) or (ii). However, such a mappig would ivolve additioal overhead. This sectio presets a differet approach which also carries over to more complicated situatios as discussed i [Qtappl]. Istead of choosig U as the W-H trasform, as i (i) & (ii), i this algorithm U is tailored to the problem uder cosideratio. The startig state γ is chose to be the specified word. The operatio U cosists of the trasfor α matio α, applied to each of the qubits ( α is a variable parameter yet to be determied) - ote that α α k k if α is, we obtai the W-H trasform of sectio. Calculatig U τγ it follows that U τγ -- --, α α = k k k this is maximized whe α is chose to be -- ; the log U. The aalysis of sectio ca k τγ = -- log log k ow be used - as i (i) & (ii), this cosists of repeatig the sequece of operatios I γ U, O times. The size of the space beig searched i this problem is! which is equal to Usig Stirlig s k k!k! approximatio: log! log, from this it follows that, comparig this to the k k k k log log log k umber of steps required by the algorithm, we fid that the umber of steps i this algorithm, as i (i) & (ii), varies approximately as the square-root of the size of the solutio space beig searched. 6

7 4. Geeral quatum mechaical algorithms The framework described i this paper ca be used to ehace the results of ay quatum mechaical algorithm. Assume there is a quatum mechaical algorithm Q due to which there is a fiite amplitude Q τγ for trasitios from the startig state γ to the target state τ. The probability of beig i the state τ is hece Q τγ - it will therefore take Ω repetitios of Q to get a sigle observatio Q τγ of state τ. Sice the quatum mechaical algorithm Q is a sequece of η elemetary uitary operatios: Q Q Q η, it is itself a uitary trasformatio. Also, Q = Q η Qη Q, i.e. Q is give by a sequece of the adjoits of the elemetary uitary operatios i the opposite order ad ca hece be sythesized. Applyig the framework of sectio, it follows that by startig with the system i the s state ad repeatig the sequece of operatios: I γ Q I τ Q, O times followed by a sigle applicatio of Q, it is possible to obtai the state τ with Q τγ certaity. 5. Sesitivity I order to achieve isolatio, quatum mechaical computers geerally have to be desiged to be microscopic - however it is extremely difficult to exert precise cotrol over microscopic idividual etities. As a result, a serious problem i implemetig quatum mechaical computers is their extreme sesitivity to perturbatios. This paper sythesizes algorithms i terms of uitary matrices - as show i sectios 3 & 4, this framework ca always be specialized to a quatum computer based o qubits; however, it ca also be applied directly to more physical situatios, hopefully reducig the eed for error correctio [Err]. For example, cosider a hypothetical implemetatio of the quatum search algorithm where the qubits are the spi states of electros ad the W-H trasform is achieved by a pulsed exteral magetic field. The results of sectio & 3 tell us that it does ot sigificatly alter the workig of the algorithm if the axes of the magets, or the periods of the pulse are slightly perturbed from what is required for the W-H trasform. Ay uitary trasform, U, close to the W-H trasform, will work provided both U & U are cosistetly applied as specified. 6. Limitatio As show i [BBHT], it is possible to express several importat computer sciece problems i such a way so that a quatum computer could solve them efficietly by a exhaustive search. Eve i physics, several importat problems ca be looked upo as searches of domais. May spectroscopic aalyses are essetially searches - a rather dramatic example of a recet search was that for the top quark. The framework of this paper could equally well be used here. All that is eeded is a meas to repeatedly apply a specified Hamiltoia that produces various phase iversios ad state trasitios. For example, it took about 0 repetitios of a certai experimet, cosistig 7

8 of iteractig a proto ad atiproto at high eergies, to obtai observatios of the top quark [Quark]. Deotig the desired state with the top quark by τ ad the iitial proto-atiproto state by γ, it implies that U τγ is approximately 0. Therefore if it were possible to apply the operatio I γ U repetitively m times, it would boost the success probability by approximately m (assumig m to be a small umber), ad it would take m fewer experimets; i case it were possible to apply the operatios I γ U, about a millio times, oe could achieve success i a sigle experimet! I priciple it is possible to sythesize U for ay uitary operatio, U, sice the adjoit of a uitary operator is uitary ad ca hece be sythesized quatum mechaically. For cotrolled operatios o qubits, sythesizig the adjoit is o harder tha sythesizig the origial operatio as discussed i sectio 4. However, the adjoit of the time evolutio operatio is the reversed evolutio operatio - this may ot be easy to sythesize whe the states are o-degeerate ad there is sigificat time evolutio. This is especially true if the time-evolutio is due to the iteral dyamics of the system. That is the mai reaso this procedure, at least i its preset form, could ot be used to isolate the top quark! This paper shows that the W-H trasform of search based algorithms ca be replaced by ay uitary operatio, provided the selective iversio operatio is carried out precisely. As metioed at the ed of sectio, eve the selective iversio ca be replaced by a selective phase shift, provided it oly affected the cocered states. Also, as metioed i sectio, the aalysis stays virtually the same if I τ is replaced by V I τ V with V uitary. Aother limitatio is that the framework of sectio demads that U & U stay the same at all time steps. What happes if there are small perturbatios i these? It seems plausible that these will ot create much impact if they are small ad average out to zero; however, that is somethig still to be proved. 7. Coclusio Desigig a useful quatum computer has bee a dautig task for two reasos. First, because the physics to implemet this is differet from what most kow devices use ad so it is ot clear what its structure should be like. The secod reaso is that oce such a computer is built, few applicatios for this are kow where it will have a clear advatage over existig computers. This paper has give a geeral framework for the sythesis of a category of algorithms where the quatum computer would have a advatage. It is expected that this formalism will also be useful i the physical desig of quatum computers, sice it demostrates that quatum algorithms ca be implemeted through geeral properties of uitary trasformatios ad ca thus adapt to available techology. The advatage of the bit represetatio is that the umber of states that ca be represeted is expoetial i the umber of bits. After the success of classical digital computers, most quatum mechaical schemes, based them- 8

9 selves o qubits. It was show that it was possible, i priciple, to sythesize gates that operate o qubits [Bit]. However idepedet qubits are ot such a atural etity i the real quatum world as bits are i the classical world. They are very delicate etities ad the slightest perturbatio affects the etire computatio. The distictive feature of this paper is that it is based o geeral properties of uitary trasforms ad ca hece be applied to a variety of situatios icludig, but ot limited to, qubit operatios. 8. Ackowledgmets Thaks to Norm Margolus & Charlie Beett for spedig the time ad effort to commet o several versios of this paper. 9. Refereces [BBHT] M. Boyer, G. Brassard, P. Hoyer & A. Tapp, Tight bouds o quatum searchig, Proc., PhysComp 996 (lal e-prit quat-ph/ ). [BBBV] C. H. Beett, E. Berstei, G. Brassard & U.Vazirai, Stregths ad weakesses of quatum computig, SIAM Joural o Computig, pp , vol. 6, o. 5, Oct 997. [Bit] S. Lloyd, A potetially realizable quatum computer, Sciece, vol. 6, pp , 993. [DJ] D. Deutsch & R. Josza, Rapid solutio of problems by quatum computatio, Proc. Royal Society of Lodo, A400, 99, pp [Err] P. W. Shor, Fault-tolerat quatum computatio, Proc. 35th Aual Symposium o Fudametals of Computer Sciece (FOCS), 996. [Factor] P. Shor, Algorithms for quatum computatio: discrete logarithms ad factorig, Proceedigs, 35th Aual Symposium o Fudametals of Computer Sciece (FOCS), 994, pp [Neural] J. Hopfield, Neural etworks & physical systems with emerget collective computatioal abilities Proceedigs Natioal Academy of Scieces, vol. 79, p. 554, 98. [Qtappl] L. Grover, A framework for fast quatum mechaical algorithms, lal e-prit quat-ph/ [Quark] T. M. Liss & P. L. Tipto, The discovery of the top quark, Scietific America, Sep. 97, pp [Reversible] C. H. Beett, Space-Time trade-offs i reversible computig, SIAM Joural of Computig, vol. 8, pages , 989. [Search] L. K. Grover, Quatum Mechaics helps i searchig for a eedle i a haystack, Phys. Rev. Letters, vol. 78(), 997, pages (lal e-prit quat-ph/ ). 9