Knight, PL

Roldan, E

Sipe, JE

Physical Review A

arXiv

Spiral - Imperial College Digital Repository

RAPID COMMUNICATIONSPHYSICAL REVIEW A 68, 020301~R! ~2003!Quantum walk on the line as an interference phenomenonPeter L. Knight,* Eugenio Rolda´n,† and J. E. Sipe‡Optics Section, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom~Received 30 April 2003; published 27 August 2003!We show that the coined quantum walk on a line can be understood as an interference phenomenon, can beclassically implemented, and indeed already has been. The walk is essentially two independent walks associ-ated with the different coin sides, coupled only at initiation. There is a simple analogy between the evolutionof walker positions and the propagation of light in a dispersive optical fiber.DOI: 10.1103/PhysRevA.68.020301 PACS number~s!: 03.67.Lx, 05.40.FbesebeFee-thahoa-itr tterponthri-ssedthtingta-llylyithnotcepa-edissssissThe quantum random walk~QW! was first proposed tenyears ago by Aharonov, Davidovich, and Zagury@1# as thequantum analog of the classical random walk~RW!. QWsare receiving much attention@2–14#: as some problems arbest solved in classical computation with algorithms baon RWs, it is expected that this type of problems couldsolved even faster in a quantum computer. Preliminary invtigations focused on the nature of the QWs themselves.example, Kempe@4# has shown that the hitting time of thdiscrete QW from one corner of anN-bit hypercube to theopposite corner is polynomial in the number of steps,n,while it is exponential inn in the classical case. Subsquently Shenvi, Kempe, and Whaley@5# showed that a QWcan perform the same tasks as Grover’s search algoriand Childset al. @6# introduced an algorithm for crossingspecial graph exponentially faster that can be done witclassical RW. Kempe@14# has recently reviewed the field.In the classical RW on the line, the ‘‘walker’’~the particleor system performing the RW! randomly takes one step tthe right or to the left depending on the result of tossingcoin. After n steps, the probability of finding the walker atdistancem from the origin is given by the binomial distribution, a Gaussian for largen with a standard deviations5An. In the QW, the role of the coin is played by a qub~as, e.g., a two-level atom or a spin-12 particle!. As its classi-cal counterpart, the quantum walker moves to the right othe left depending on the internal state of the qubit. Afeach displacement, the state of the qubit is set to a supesition state by means of a suitable unitary transformatitypically a Hadamard, that plays the role of the toss ofcoin in the RW. Yet the QW is not arandomwalk, as its timeevolution is completely deterministic. The probability distbution in the quantum case is very different from the clascal one: it resembles the Airy function~Fig. 1! and has astandard deviation that is linear withn. This is thediscretetime QW that should be distinguished from thecontinuoustime QW @2,14#, which we will not consider here.*Email address: p.knight@imperial.ac.uk†Permanent address: Departament d’O` ptica, Universitat de Vale`n-cia, Dr. Moliner 50, 46100-Burjassot, Spain. Email addreeugenio.roldan@uv.es‡Permanent address: Department of Physics, Univerof Toronto, Toronto M5S 1A7, Canada. Email addresipe@physics.utoronto.ca1050-2947/2003/68~2!/020301~4!/$20.00 68 0203des-orm,aaoro-,ei-Possible implementations of the QW have been propoby a number of authors@10–12#. Here we show that aclas-sical implementation of the QW is possible, in analogy wiother processes usually associated with quantum compu@15–18#. Indeed, we point out that a classical implemention very similar to the one we are proposing has actuabeen implemented by Bouwmeesteret al. @19#, in the contextof the optical Galton board, without the authors explicitnoting this. Other classical~interferometric! implementationsof the QW have been proposed recently@8,13#, but in themthe number of necessary optical elements grows quickly wthe number of steps in the QW, something that doesoccur in our scheme. Finally, by reexamining the differenequations for the walker we show that the nature of progation is simpler than has been previously appreciated.In our classical approach the role of the walker is playby the frequency of a light field, and the role of the coinplayed by its polarization state. The light fieldEW 5 (m52 llEW m exp@ i ~v01mv̄ !t2 ikmz#1c.c. ~1!:ty:FIG. 1. ~a! Probability distribution forn5200 for both the clas-sical ~dashed! and quantum~continuous! random walks. The initialconditions chosen for calculating the QW wereR0,051/& andL0,05 i /&, see Eqs.~4! and~5!. Notice that the quantumPm is nullfor odd m at oddn. We have represented only nonzero values.~b!Continuous limit of the QW as given by Eq.~13! for a50.4 andt5200 with the same initial conditions as in~a!.©2003 The American Physical Society01-1anrathsheat-Thuenthtdlightnitydcedre.callyondams intalre-be-isans-llyiftofencetothe.iednssitlyeirdy-teslks,a,nev-RAPID COMMUNICATIONSKNIGHT, ROLDÁN, AND SIPE PHYSICAL REVIEW A68, 020301~R! ~2003!(v0 is the carrier frequency andv̄ is the frequency differ-ence between successive frequency components! can be rep-resented by the abstract stateuc)5 (m52 ll@Rmum,x!1Lmum,y)], ~2!where Rm[ x̂•uW m and Lm[ ŷ•uW m (uW m5EW m /uEW mu) andSm52 ll @ uRmu21uLmu2#51; the ‘‘basis vectors’’um,c! labelthe frequency and polarization, withc5x,y; we associatedx(y) with the coin head~tail!.To implement the walk, we require a unitary operator thperformsV̂um, yx)5um61, yx).The operationV̂ can be physically implemented, e.g., with aelectrooptic modulator~EOM! to which a linearly time de-pendent voltage is applied in such a way that thex ~y! polar-ization component of the field frequency component (v01mv̄) will see its frequency increased~ ecreased! by anamountv̄.After each jump in the frequency of the field, a HadamatransformationĤum, yx)51&@ um,x!6um,y)]has to be implemented. This can be done optically by meof a half-wave plate~HWP! with its fast axis forming ananglep/8 with respect to thex̂ axis @15,16#. Finally, the QWis implemented by the repeated action on the state ofoperatorĤV̂, i.e., aftern iterations zc(n)…5@ĤV̂#nzc(0)…,which can be written aszc~n!…5 (m52n1n@Rm,num,x!1Lm,num,y)], ~3!Rm,n51&~Rm21,n211Lm11,n21!, ~4!Lm,n51&~Rm21,n212Lm11,n21!, ~5!whereRm,05Lm,050 if mÞ0 andRm,215Lm,2150 ;m.These are the standard QW equations. Finally, the intenof each frequency component of the light field, which is toptical analog of the probability of finding the walkerposition m at iteration~time! n, is given by Pm,n5uRm,nu21uLm,nu2, which is represented in Fig. 1.In order to implementn steps the best option is to introduce the described elements in an optical cavity, Fig. 2.cavity imposes a constraint that the optical frequencies mfit within its set of eigenfrequencies. Thus, the time depdent electric field applied to the EOM and the cavity lengmust be adjusted in such a way that the frequency shifv̄5 f vFSR with vFSR being the cavity free spectral range anf02030tdnseityest-being an integer number. Consider nevertheless that apulse with a spectral widthDv is initially injected in thecavity. Although in such a casef does not need to be ainteger, in order to perform a step of the QW at each cavroundtrip, the step sizev̄ must be large enough to avoisignificant overlap between the spectra of the displapulses; thus the frequency steps are well resolved.The experiment of Bouwmeestert al. @19# can be seen asa realization of the QW very similar to that proposed heThese researchers proposed and studied, both theoretiand experimentally, an optical implementation of the GaltBoard~the quincunx!. What they actually implement is a griof Landau-Zener crossings through which a light bepropagates, and concentrate on the study of recurrencethe light spectrum. A simplified version of their experimendevice is that represented in Fig. 2, but with the QWPplaced by a second EOM with its axis rotatedp/4 with re-spect to the first EOM, which introduces a dephasingtween the two polarization components. Although thunitary operation does not correspond to a Hadamard trformation, it can be shown that it leads to an essentiaidentical QW @5# ~details to be reported elsewhere!. Themain difference with our proposal is that the frequency shintroduced by the EOM ismaller thanvFSR and then eachstep in the QW takes several cavity round-trips. In Fig. 6Ref. @19# the QW is clearly seen. Bouwmeestert al. @19#considered this case as a demonstration of the coherquality of their system, and did not note its significanceQWs; their focus on the observation of recurrences inspectrum led them to study other aspects of their systemLet us now reexamine the linear difference equations~4!and ~5!. They admit a formal solution that has been studfrom a number of points of view, usually with a focus oidentifying its asymptotic behavior for largen @3,9,14# as itallows for the extraction of much information. Neverthelethe formal solutions presented to date do not rely explicon a crucial feature of Eqs.~4! and ~5!, which we now ex-plain, that greatly simplifies a physical understanding of thsolution. A little algebra reveals that the solutionsRm,n andLm,n of Eqs.~4! and ~5! also satisfyam,n115am,n2111&@am21,n2am11,n#, a5R,L. ~6!This is a remarkable equation, since it demonstrates anamical independence of the evolution of the two coin staR and L. Thus there are two essentially independent waFIG. 2. Scheme for the optical implementation of the QW inFabry-Perot cavity. The electro-optic modulator~EOM! shifts thefield frequency up or down inv̄/2 depending on its polarizationand a quarter-wave plate~QWP! with its axis forming an anglep/8,with respect to thex axis, performs the Hadamard transformatio~notice that light passes twice through each intracavity elementery round-trip!.1-2acvetenut.-e-nionsin.on-anhthen ais-via-ga-nein-g-tedapsionicleid-tera-ner-ise isanoithRAPID COMMUNICATIONSQUANTUM WALK ON THE LINE AS AN INTERFERENCE . . . PHYSICAL REVIEW A 68, 020301~R! ~2003!coupled only by the first step that linksam,1 to am,0 . Afterthat, the two walks can be studied independently of eother.The most naı¨ve continuous limit of Eq.~6! would involvea first derivative with respect to time and a first derivatiwith respect to space, and would suggest waves propagaonly towards1` for both R andL, in apparent violation ofthe symmetry of the problem. But this is too simplistic, givthat for bothR andL one can look for solutions of the formam,n5Am,n1 1~21!nAm,n2 , ~7!whereAm6 satisfyAm,n116 2Am,n216 561&@Am21,n6 2Am11,n6 #, ~8!restoring the symmetry. Of course, there is not a uniqspecification of theAm6 in terms of the fundamentalam ,sinceam,05Am,01 1Am,02 , ~9!am,15Am,11 2Am,12 .The specification of theam,0 and am,1 , which completelyspecifies the initial conditions required to solve Eq.~6!, doesnot suffice to determine the initial conditionsAm,06 andAm,16required for the solution of Eq.~8! uniquely. Nonetheless, iis possible to rigorously develop the solution of Eqs.~6! interms of the fieldsAm,n6 ; this we defer to a later publicationThe point we wish to stress here is that in the limit ofAm,n6that are slowly varying inn andm we can introduce continuous functionsA6(x,t) and understand Eq.~8! as the discreti-zation of the differential equation(k50`~Dt !2k11~2k11!!]2k11]t2k11A6~x,t !571&(k50`~Dx!2k11~2k11!!]2k11]x2k11A6~x,t !, ~10!where Dt and Dx denote the temporal and spatial incrments, respectively. Keeping only the first two terms aapproximating the third derivative in time using the equatat the lowest order, we obtain]]tA6~j,t!571&F ]]j 1 112 ]3]j3GA6~j,t!, ~11!wheret5t/Dt andj5x/Dx. In this slowly-varying approxi-mation conditions~9! can be approximated asA6„m~Dx!,0…'Am,06 5 12 ~am,06am,1!, ~12!where we have made use of Eq.~7! and assumed thatam,15Am,11 2Am,12 'Am,01 2Am,02 . Thus in this limit Eq.~12! pro-02030hingedvides the initial conditions for Eq.~11! and those equationcan be solved by Fourier analysis. There are thentwo fieldsA6(j,t) that can be associated with each side of the coThis feature persists when the rigorous solution is cstructed in this terminology, where there is a~temporal! ‘‘fer-romagnetic’’ field Am,n1 and an ‘‘antiferromagnetic’’ field(21)nAm,n2 for each coin side.Returning to Eq.~11! we make use of Eqs.~12! and takeas initial conditions A6(j,0)5a0,0G(0)6a21,1G(21)6a1,1G(1), with G(j0)5N exp@2(j2j0)2/(2a)2# and N anormalization factor; here we assume that Eq.~11! is onlycorrect for the long-wavelength components by takinginitial condition that ‘‘smears out’’ the lower-wavelengtcomponents. The solution is easily found analytically@20,21#for A6(j,t), and we can write the final result~up to a nor-malization factor! for both R and L as a(j,t)5A1(j,t)1(21)nA2(j,t), withA6~j,t!5a0,0Z~6j,t!6a21,1Z„6~j1j0!,t…6a1,1Z„6~j2j0!,t…, ~13!Z~j,t!52pB1/3expS 3ABC12C33B2 DAi S AB1C2B4/3 D , ~14!where A5j2t/&, B5t/(4/&), C5a2 and Ai(x) is theAiry function @21#; the R and L solutions differ only in thedifferent values ofam,0 andam,1 appearing in Eq.~13!. Theappearance of Airy functions in the full solutions of Eqs.~4!and ~5! @9# can thus be understood as associated withform of Eq. ~11!, which workers in fiber optics will recog-nize as the classical equation for the propagation of light ifiber with no group velocity dispersion but a third-order dpersion term. The linear dependence of the standard detion on n arises, of course, simply because of this propation. Solution~13! is represented in Fig. 1~b! for a50.4, andthe similarity with the QW in Fig. 1~a! is clearly apparent.In conclusion, we have shown that the QW along a lican be simulated in a purely classical implementation,volving nothing more than wave interference of electromanetic fields. And, indeed, it has in fact already been simulain the laboratory in the work of Bouwmeestert al. @19#.Further, this classical nature of the propagation is perhnot surprising. After all, the standard QW is a generalizatof the quantum-mechanical problem of a spinless partwith hopping amplitudes between sites, familiar from solstate physics if the time variable is continuous. That latproblem, which gives a simple Schro¨dinger equation in itscontinuum limit, is clearly classical in nature of its propagtion, as attested to by the appearance of the Schro¨dingerequation in classical beam propagation problems. The gealization involved in concocting the standard QW problemthe inclusion of a spin variable. What we have shown herthat this generalization does not affect the dynamics inessential way. Except for an initial coupling in the first twtime steps, the evolutions of the amplitudes associated wthe two sides of the coin proceed independently.1-3ngpeisv--ringgeRAPID COMMUNICATIONSKNIGHT, ROLDÁN, AND SIPE PHYSICAL REVIEW A68, 020301~R! ~2003!This work was supported in part by the UK Engineeriand Physical Sciences Research Council and the EuroUnion. E.R. acknowledges financial support from the Minterio de Educacio´n, Cultura y Deportes of the Spanish Gondtntel,02030an-ernment~Grant No. PR20002-0244!. J.E.S. acknowledges financial support from the Natural Sciences and EngineeResearch Council of Canada. We gratefully acknowledfruitful discussions with V. Kendon.ht,nt.. J.J.@1# Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev. A48,1687 ~1993!.@2# E. Farhi and S. Gutmann, Phys. Rev. A58, 915 ~1998!.@3# A. Nayak and A. Vishwanath, e-print quant-ph/0010117.@4# J. 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Quantum walk on the line as an interference phenomenon

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