A Generalization of the Deutsch-Jozsa Algorithm to Multi-Valued Quantum Logic

Transcription

1 A Geealizatio of the Deutsch-Jozsa Algoithm to Multi-Valued Quatum Logic Yale Fa The Catli Gabel School 885 SW Baes Road Potlad, OR , USA Abstact We geealize the biay Deutsch-Jozsa algoithm to - valued logic usig the quatum Fouie tasfom. Ou algoithm is ot oly able to distiguish betwee costat ad balaced Boolea fuctios i a sigle quey, but ca also fid closed expessios fo classes of affie fuctios i quatum oacles, accuate to a costat tem. Itoductio The oigial biay Deutsch-Jozsa algoithm [] cosides a Boolea fuctio of the fom f : {, } {, } implemeted i a black box cicuit, o oacle, U f. Iput states ae put i a quatum supepositio as quey (x) ad aswe (y) egistes so that thei state vectos ae expessed i tems of the dual basis [3] = ( + ) ad = ( ) The oacle is defied by its actio o the egistes U f xy = x y f(x), whee the x egiste is the teso poduct of iput states x x. Whe it is pomised that the fuctio i questio is eithe costat (etuig a fixed value) o balaced (etuig outputs equally amog ad ), the algoithm decides detemiistically which type it is with a sigle oacle quey as opposed to the + equied classically. The coespodig cicuit is show below (/ deotes wies i paallel). I this pape, we pove a extesio of the Deutsch- Jozsa algoithm to abitay adices of multi-valued quatum logic. We deote additio ove the additive goup Z by the opeato ad the Koecke teso poduct by. The Hadamad tasfom is a special case of the quatum Fouie tasfom (QFT) i Hilbet space H. The wellkow Chesteso gate fo teay quatum computig is x : / H Uf y : H x H y f(x) H ψ ψ ψ 3 ψ 4 Figue. The Deutsch-Jozsa cicuit also equivalet to the Fouie tasfom ove Z 3. Ceeceda [] has geealized the Deutsch algoithm usig two qudits fo d-dimesioal quatum systems whee d = k. Howeve, placig o estictios o the umbe of computatioal basis states leads to a fa moe vesatile chaacteizatio of the Deutsch-Jozsa algoithm. The Fouie matix of ode ove the pimitive th oots of uity k = e iπk/ is give i Figue. F = 4 ( ) 3 6 3( ) ( ) ( )( ) Figue. The QFT as a matix Defiitio The actio of the quatum Fouie tasfom is descibed by QFT : j e iπjk/ k fo j k= Z [3]. Fo ease of computatio, the QFT ca be expessed

2 as F = e iπjk/ j k j= k= Lemma Pimitive th oots of uity satisfy αk = fo ozeo iteges α. k= The -ay quatum Fouie tasfom gives omalized otatios of a vecto, poducig supepositios that diffe oly i phase. This leads to a edefiitio of the dual basis fo H as {,,..., } = { } x, x x,..., ( )x x The -ay Deutsch-Jozsa Algoithm ad Affie Fuctios We will implemet the QFT as a geealizatio of the Hadamad tasfom i the multi-valued equivalet of the Deutsch-Jozsa algoithm. Fist, we cove some extesios by defiitio. Defiitio A -qudit multi-valued fuctio of the fom f : {,,..., } {,,..., } is costat whe f(x) = f(y) x, y {,,..., } ad is balaced whe a equal umbe of the domai values, amely, is mapped to each of the elemets i the codomai. I multi-valued logic, thee ae costat fuctios mappig each elemet i Z to a fixed elemet ad! balaced pemutative (bijective) mappigs of sigle-qudit iputs. Fo fuctios o qudits, thee ae accodigly!/! balaced (sujective) mappigs. Theoem All affie fuctios defied as f(x,..., x ) = A A x A x with A,..., A Z ae eithe costat o balaced fuctios of qudits. Poof. Those affie fuctios fo which all coefficiets A i = ae costat. Fo affie fuctios with at least oe ozeo coefficiet of x i, each elemet i the domai {,,..., } = {,,..., } is educible modulo to a uique elemet m of Z because the domai is equivalet to the set {m, m, m,..., +m} of size, all of whose membes ae coguet to m modulo fo evey m {,,..., }, a set of size. Sice f(p) = f(q) if p q (mod ), evey elemet i the codomai {,,..., } is assiged to exactly diffeet elemets i the domai. Such affie fuctios satisfy the defiitio of a balaced fuctio. The poof of the -ay Deutsch-Jozsa algoithm will be aided by a tivial lemma: Poof. Coside the polyomial z =, of which α is a oot. beig a eal oot fo all iteges, this ca be factoized as (z )(z +z + +) =. Theefoe, k= zk = fo z = α whee z =. This leads to ou mai esult. Theoem The -ay Deutsch-Jozsa algoithm applied to multi-valued fuctios of qudits ca both distiguish betwee costat ad balaced fuctios with a sigle oacle quey ad detemie a closed expessio fo a affie fuctio i U f, exceptig the costat tem, as follows:. The costat tem A is peseved i the phase of the x-egiste at output ( A ), which is lost duig measuemet.. The coefficiets A,..., A ae detemied by the state of the x-egiste at output, A,..., A. x : / F U f F y : F F ψ ψ ψ 3 ψ 4 Figue 3. Cicuit fo the -ay Deutsch-Jozsa algoithm I pactice, the y-egiste would ot be measued, but we follow though with the calculatios fo it to demostate that its state at the output is costat, egadless of the fuctio i the oacle. The x- ad y-egistes ae witte sepaately as factos of the etie tesoed state of the cicuit at each step ψ i. Fist, we coside the case i which the fuctio f(x) hidde i the oacle is costat: ψ = F + ψ = x e iπy/ y y= y= U f ψ3 = x e iπy/ y f(x)

3 At this poit, we ca tasfe the actio of U f fom the basis states themselves oto thei phases by obsevig that if a basis vecto j k is appeded with the phase φ j, the j itself must have phase φ j k by defiitio. This yields: ψ 3 = x e iπ[y f(x)]/ y y= = e iπf(x)/ x e iπy/ y y= Because we assume ou fuctio to be costat, e iπf(x)/ ca be egaded as a global phase facto. Subsequetly, the QFT o the x-egiste ca be computed explicitly: F = e iπf(x)/ j= k= F + ψ 4 = j= k= e iπjk/ j k, givig e iπjk/ j k x e iπ(jk y)/ j k y j= k= y= I the stadad basis, k z = whe k z, while k z = othewise. We ca hece educe the above to: ψ 4 = e iπf(x)/ j= k= e iπ(j )k/ j j= k= e iπjk/ j By lemma, all basis states j i the x-egiste will have ull amplitudes fo j. Similaly, all basis states j i the y-egiste will have ull amplitudes fo j. It follows that ψ 4 = with a phase facto of e iπf(x)/ fo all costat fuctios f(x). The balaced case is simila. Afte iitializig ad supeposig ou states as above, we obtai: ψ 3 = e iπf(x)/ x e iπy/ y y= I this case, the phase facto e iπf(x)/ caot be assumed to be global because its value is depedet upo x. The output of the y-egiste will be the same as i the costat case, so we eed oly to poceed with the state of the x- egiste. Afte applyig the secod QFT: = j= = j= k= ψ 3 F ψ 4 e iπjk/ e iπf(x)/ j k x e iπ[jx f(x)]/ j It is ow ecessay to show that jx f(x) = some costat C, o f(x) = jx C, fo a fixed value of j ad all x i domai {,,..., }. This would allow e iπc/, o C, to be the phase facto of some basis state j othe tha as i the costat case, with a detemiistic pobability of measuemet. Equivaletly, sice additio ad multiplicatio ae modula, f(x,..., x ) = C j x j x must hold fo some j {,..., }, j i epesetig the i th digit of j. This is the defiitio of a o-costat affie fuctio, so j exists oly whe the hidde fuctio i the oacle is affie. The x-egiste is theefoe measued to be: ψ 4 = e iπ[jx f(x)]/ j = C j, f(x) beig balaced fo j ad costat othewise. I cosequece, the Deutsch-Jozsa algoithm gives a detemiistic output oly whe f(x) is esticted to beig eithe costat o balaced, ad affie (we will give a example of the algoithm fo a o-affie fuctio below). Howeve, obseve that the state will have a zeo pobability of measuemet fo all balaced fuctios, whethe affie o ot, because its amplitude is e iπf(x)/ e iπf(x)/ e iπx/ = by defiitio ad lemma. Thus, the algoithm is still detemiistic i the sese that it ca always distiguish betwee costat ad eithe affie o o-affie balaced fuctios, although with o fixed output i the latte case. Fially, the geealized Deutsch-Jozsa algoithm has a useful popety that becomes appaet i multi-valued logic; it ca ot oly distiguish betwee costat ad balaced fuctios, but ca detemie explicitly the fuctio f(x,..., x ) = A A x A x implemeted by the oacle exceptig the costat tem A, give that it is 3

4 affie. As calculated above, the costat tem A of such a fuctio is ecoded i the phase of the x-egiste at output (A = C, whee the phase is C ), while the espective coefficiets A,..., A of x ae detemied by the basis vecto j = A,..., A. Sice the phase of the x-egiste is lost at measuemet, oly A caot be etieved. We egad affie fuctios that diffe oly i the costat tem as a class. I some vaiatios of the Deutsch-Jozsa algoithm [5], the y-egiste is uecessay if the oacle, coespodig to the diagoal opeato U f = e iπf(x)/ x x, diectly ecodes the actio of f(x) ito the phase of the x- egiste. We use this scheme below. Example (Deutsch-Jozsa fo a affie fuctio) U f cotais the followig balaced fuctio defied o two qutits AB : B A We begi at ψ 3, afte the states have bee iitialized. ψ 3 = 3 = 3 Lemma is used fo simplificatio ( + + = ): 9 F ψ 3 = ψ 4 =, = fom which we deive a closed expessio fo the affie fuctio f(x, x ) = A A x A x i U f, save fo the costat tem, by takig {, } as the espective coefficiets A ad A (although theoetically, A should be ( ) mod 3 = ): f(x, x ) = (Costat) x x Example (Deutsch-Jozsa fo a o-affie fuctio) U f cotais the followig balaced fuctio defied o two qutits AB : Agai begiig at ψ 3 : B A ψ 3 = 3 = 3 9 F ψ 3 = = ψ 4 = /3 (+ )/3 /3 /3 /3 (+)/3 The basis state hece has a 4/9 pobability of measuemet, with all othes havig /9 pobability. To detemie that this balaced fuctio is o-affie, eough measuemets of the x-egiste ae equied so as to obtai diffeet states. Futhemoe, obseve that if the fuctio is ot affie, thee is still a elatively high pobability of measuig the output state of the algoithm that would be associated with a simila affie fuctio. Fo example, compae the Maquad chat of the o-affie fuctio i U f above with that of the affie fuctio f(x, x ) = x x associated with the output state : x x Because a affie multi-valued fuctio i ou cotext is defied i tems of the modulo-additive opeato, ay abitay fuctio is affie iff it satisfies the cyclic goup popety (this is easy to detemie whe the outputs ae plotted 4

5 i a Maquad chat as above, which equies that ows ad colums be successive cyclic shifts of each othe), ad the state afte U f is factoable - e.g., etaglemet does ot occu i the Deutsch-Jozsa cicuit. Ou exteded Deutsch-Jozsa algoithm equies oly oe measuemet to detemiistically distiguish a expessio fo a affie fuctio of adix ad iputs up to the accuacy of a costat, give that it is affie. 3 Coclusio Although the oigial Deutsch-Jozsa algoithm is maily of theoetical iteest, its multi-valued extesio could potetially fid applicatio i image pocessig to distiguish betwee maps of textue images ecoded by affie fuctios i a Maquad chat, with the umbe of colos coespodig to the size of the adix. Ackowledgemets The autho gatefully ackowledges Pofesso Maek Pekowski of Potlad State Uivesity Depatmet of Electical ad Compute Egieeig fo his guidace ad Jacob Biamote of D-Wave Systems, as well as the membes of the Potlad Quatum Logic Goup fo thei suppot. Appedix Fo adices highe tha i which U f is meely + wies i paallel (implemetig the costat fuctio f : {, } ), the iitialized state is mapped to a diffeet output. Pactically, this makes o diffeece because the output of the y-egiste is discaded. Howeve, theoem 3 makes clea why this is so. Theoem 3 Fou iteatios of the QFT gives a idetity mappig (the biay Hadamad gate is a special case that is also self-ivese). Poof. Let [F ] pq deote the p, q ety i F. Row p ad colum q of F ae give by p = k(p ) k ad q = k(q ) k, k= k= espectively. [F] pq = p q = k= k(p+q ) ; p+q is some itege α that is ozeo whe (p+q) mod, so by lemma, [F] pq = if (p + q) mod = ad [F] pq = othewise. By defiitio, all such idices p, q {,,..., } fo which [F] pq = satisfy p+q = C + fo some C Z. C is futhe esticted to {, } because ay lage values of C + exceed the maximum value of p + q, o. Theefoe, eithe p + q = o p + q = +, givig the solutio sets p = q = o {p, q} = { m +, m + } fo m {,,..., }, which coespod to the pemutatio matix F = m= m m, take modulo to mea Thus, (z deotig the complex cojugate) F 3 = = = = F j= k= m= e iπjk/ m m j k e iπjk/ j k j= k= (e iπjk/ ) j k j= k= by ou pevious agumets, ad give that F is uitay, F 4 = F F = I is immediate. This is a popety that the QFT shaes with the cotiuous Fouie tasfom. Refeeces [] Deutsch, D. ad Jozsa, R. (99). Rapid solutios of poblems by quatum computatio. Poc. Roy. Soc. Lod. A 439, [] Ceeceda, J. L. (4). Geealizatio of the Deutsch algoithm usig two qudits. Available at quatph/4753. [3] Jozsa, R. (997). Quatum algoithms ad the Fouie tasfom. Poc. Roy. Soc. Lod. A 454, [4] Pekowski, M. (5). Lectue otes. Available at mpekows/semia.html. [5] Guska, J. (999). Quatum computig. Bekshie, UK: McGaw-Hill. [6] Pey, R. T. (6). The temple of quatum computig, vesio.. Available at 5