Discrete-time quantum walk with two-step memory

Transcription

1 Discrete-time quatum walk with two-step memory Christopher Joes Departmet of Physics Orego State Uiversity Faculty Project Advisor: Yevgeiy Kovchegov OSU Mathematics Advisor: Zlatko Dimcovic OSU Physics Jue 7, 01 Abstract We examie a discrete-time quatum walk with two-step memory for a particle o a oedimesioal ifiite space. The walk is defied with a four-state memory space aalogous to the two-state coi space commoly used i discrete time quatum walks, ad a method is preseted for calculatig the time evolutio by usig the Fourier trasform. A itegral expressio for the probability is calculated, ad this is used to produce umerical solutios for the probability distributio as a fuctio of the time step ad positio. The results show two peaks i the probability distributio. Oe peak propagates ballistically with time, which is a commo feature of quatum walks. The other peak is statioary with time ad located at the iitial site of the particle. This feature is ot commo i quatum walks ad suggests that tracig the immediate history of the particle usig two-step memory may represet the begiig of a trasitio to a classical system. Fuded by URISC 1

2 Cotets List of Figures 1 Itroductio Quatum Walks Quatum Walks With Memory Methods 8.1 Quatum Walk o a Ifiite Positio Space Types of Tails Evolutio Operator Fourier Trasform Diagoalizig the Operator M Iverse Fourier Trasform Results Tail Probability Amplitudes Probability Distributio Discussio Tail Probability Amplitudes Probability Distributio Coclusio 3 6 Ackowledgemets 5 7 Refereces 6 List of Figures 1 Probability amplitude a t () as a fuctio of positio for time steps 99, 100, 199, Probability amplitude b t () as a fuctio of positio for time steps 99, 100, 199, Probability amplitude c t () as a fuctio of positio for time steps 99, 100, 199, Probability amplitude d t () as a fuctio of positio for time steps 99, 100, 199, Probability distributio P t () for the particle as a fuctio of positio for time steps 99, 100, 199,

3 1 Itroductio Whe itroducig the idea of quatum radom walks, it is usually helpful to begi with a brief descriptio of what a Walk is. Quatum walks were motivated by applyig aspects of classical walks to quatum systems [1]. I the classical walk, the particle is located at oly oe positio ad at each step i time it moves i oe of several possible directios. The particle is free to take discrete steps, or walk, aroud the set of coected positios. Classical walks ca be studied o may differet positio spaces, however the simplest to cosider is the oe-dimesioal walk. This will be especially relevat because the quatum walk discussed i this paper is also a oe dimesioal walk. Cosider a classical particle that is free to move o a ifiite oe-dimesioal space of discrete positios. At each step i time the particle ca move i oe of two directios, say left or right. The probability that the particle moves oe way or aother depeds o the dyamics of the system, but the most commo example is the symmetric walk i which steps to the left ad right are equally probable. Oe ca thik of this as flippig a coi at each time step ad movig the particle to the left or right depedig o whether the coi comes up heads or tails. Sice there is equal probability for steps to the left ad right, after a large amout of time oe would expect there to be roughly equal umbers of steps i each directio. Overall, these steps will cacel out ad the particle would be expected to stay i the area aroud where it started. The result is a approximately Gaussia probability distributio cetered o the iitial positio of the particle []. This example is oe of the simplest possible walks that ca be cosidered, but it is a good illustratio of the basic priciples. More sophisticated statistical aalysis is required to solve more complicated walks, but the goal is usually the same: fid the spatial probability distributio for the particle at some later time. Classical walks ca provide good models to represet physical systems, for example, diffusio of particles []. I additio, the study of walks is also very importat i computer sciece, leadig to the developmet of various computer algorithms [3]. Due to the success of classical radom walks, much of the recet iterest i quatum walks is based o the belief that they will lead to deeper uderstadig i other fields. The most promisig of these is the hope that quatum walks ca be used to develop ew algorithms i quatum computig. Curretly, the two mai techiques for aalyzig quatum algorithms are Fourier samplig ad amplitude amplificatio [4]. The former is represeted i the work of Daiel Simo [5] ad Peter Shor [6], ad the latter is described i the work of Lov K. Grover [7]. Shor s factorig algorithm is able to factor large umbers i a time that oly grows polyomially (as opposed to expoetially) with the legth of the umber [6], ad Grover s search algorithm is able to search databases at a rate much faster tha ay classical algorithm [7]. However, quatum walks have recetly bee established as a alterative method for developig algorithms, ad there have bee several impressive results. For example, quatum walks have bee show to provide expoetial speedup over classical computig [8], ad algorithms have bee developed that are able to solve problems such as searchig o graphs ad checkig matrix multiplicatio [1]. With some of the possible beefits to studyig quatum walks ow outlied, the ext step is to better describe what a quatum walk is ad how it differs from the classical radom walks. 3

4 1.1 Quatum Walks Although the developmet of quatum walks was based o classical radom walks, there are some key differeces betwee the two. The first is that a classical particle ca exist at oly oe positio at a give time, but the quatum particle ca exist i a superpositio of states util a measuremet takes place. This meas that the state of the quatum particle ca iterfere at differet sites leadig to probability distributios that are much differet tha the classical distributios. A secod differece betwee quatum ad classical walks is that all time evolutio i the quatum walk must be uitary, meaig that the processes are reversible. This uitary behavior becomes very importat whe cosiderig the two differet types of quatum walks. The study of quatum walks ca be divided ito discrete-time quatum walks (DTQW) ad cotiuous-time quatum walks (CTQW), ad the two types are fudametally differet [1]. Discrete-time walks refer to evolutio that occurs betwee small quatized steps i time; at each time step the particle chages its positio. Cotiuous-time meas that the particle ca move at ay of a cotiuous rage of times ad ot just i discrete steps. This paper is focused o solvig a DTQW so there is little eed to go ito CTQWs i depth. However, there are some importat distictios that should be oted. First, to produce otrivial behavior i the DTQW, oe is required to itroduce a ew degree of freedom to the state of the particle [1]. This degree of freedom is sometimes referred to as the coi space or directioality of the particle. CTQWs, o the other had, do ot eed this extra degree of freedom. This fudametal differece betwee DTQWs ad CTQWs meas that there is ot yet a kow way to coect the two categories. I classical walks, cotiuous-time radom walks ca be cosidered a limitig case of the discrete-time walks as the time steps become very small. However, o such coectio ca be made for quatum walks. For more o the relatioship betwee DTQWs ad CTQWs, refer the article by A. M. Childs [1]. This paper focuses o solvig a oe dimesioal DTWQ, so it will help to first expad o the idea of the coi space ad how it works. For a oe-dimesioal walk, there will be two coi eigestates correspodig to the two directios that the particle ca move. For simplicity we ca just refer to them as left (L) or right (R) correspodig to the directio a particle with that state will move. The positio of the particle is a separate space ad the full state of the particle is the tesor product of the coi state ad the positio state. ψ = coi (1.1) Note that two states with differet positios could be i the same pure R state. For example, a state at positio 0 ad oe at positio 5 could both be i the same (R) coi state. ψ 1 = R 0, ψ = R 5 Also, oe particle at some positio could be i superpositio of R ad L states. For example, a state at positio 0 could be i a equal superpositio of (R) ad (L). ψ = 1 ( R + L ) 0 = 1 ( R 0 + L 0 ) Remember that the state must be ormalized, as is required by quatum mechaics. The way that the particle moves is described by the uitary evolutio operator, which icludes the coi operator ad the shift operator. The coi operator acts o the particle chagig its coi 4

5 state. The ame coi operator is i referece to the aalogy from classical walks i which a coi flip determies the directio of each step. The directio i which the particle will move depeds o whether the particle is i the R or L state. This process is carried out by a separate shift operator, ad the actio of both operators is uitary. This process of mixig the coi state ad shiftig the particle is repeated may times to evolve the overall state, ad the goal of quatum walks is to solve for the probability distributio of the particle at that later time. The most commo example of this type of walk is the Hadamard walk. I this case the coi operator, C, acts o the coi states i the followig way. C R = 1 ( R + L ) C L = 1 ( R L ) (1.) So the coi operator takes the pure coi states ito equal superpositios of each coi state. The mius sig i the secod equatio is required to make the operator uitary. If a step to the right is cosidered movig i the positive directio ad a step to the left is i the egative directio, the shift operator, S, acts i the followig way. S( R ) = R + 1 S( L ) = L 1 (1.3) The shift moves a R state to the right, icreasig the positio from to + 1, ad a L state to the left, decreasig the positio from to 1. The total evolutio operator is the combiatio of the coi ad the shift operator U = S(C I) (1.4) where I is the idetity operator. The coi operator must be applied first because it assigs the probabilities for the particle to travel i ay give directio. The evolutio of the state betwee time steps is give by ψ t+1 = U ψ t = S(C I) ψ t. (1.5) As a example, if a particle starts i the pure (R) state at positio 0, applyig the operator twice gives the ext two evolved states. ψ 0 = R 0 ψ 1 = 1 L R 1 ψ = 1 L + 1 ( R + L ) R (1.6) Fially, the probability to fid the particle at a positio is calculated usig P () = ψ. For example, the probability that the particle is foud at = 0 i ψ from Eq. (1.6) is [ P (0) = 0 1 L + 1 ( R + L ) ] R (1.7) (1.8) 5

6 P (0) = 1 ( R + L ) = 1 4 R R L L = 1. (1.9) Clearly, the state of the particle will expad rapidly as it evolves ad at times larger tha t =, the probability distributios will be much harder to calculate. A more i-depth descriptio of this walk is give i each of the refereces [1,,3,9], icludig how to express the state ad the spatial probability distributios after may time steps. 1. Quatum Walks With Memory Zlatko Dimcovic ad Yevgeiy Kovchegov developed a framework for solvig DTQW that is based o the idea of a walk with memory [9]. This meas that the state of the particle icludes ot oly its curret positio, but also a memory of its previous positio. Keepig track of the particle s previous positio ca be thought of as aalogous to the coi space itroduced i other quatum walks. The state of the particle is the give by the tesor product of the previous ad curret positio. ψ = previous curret If the positio at a time t is t, the the previous positio is t 1. So the state of the particle at the time t would be ψ t = t 1 t. (1.10) For a quatum walk o a oe-dimesioal lie, a particle at a give locatio could have come from oe of two adjacet positios. Therefore, there will be two possible memory states just like there were two coi states. For example, a particle at positio could have arrived there from 1 ad would be described by the state ket 1. Similarly, a particle that arrived from +1 would be described by + 1. The evolutio operator the acts differetly o the state depedig o where the particle came from. These memory states are still aalogous to the coi states for the walk o a lie. Cosider the state 1. Its most recet step was to the right, so this state could also be described usig the ket R, ad a direct relatio ca be made betwee the memory states ad the coi states described i the previous sectio. R 1 L + 1 (1.11) I this way, the framework for quatum walks with memory is able to exactly recreate the stadard coied walk o the oe dimesioal lie. However, this framework has prove to be more flexible tha the stadard coi approach whe solvig certai problems. For example, i their paper Dimcovic ad Kovchegov were able to solve a DTQW o a biary tree, a problem that, despite much iterest, had ot yet bee solved usig stadard coi methods [9]. I additio, this approach does ot require oe to itroduce some ew degree of freedom like the coi space. Istead, oe just eeds to exted the positio space to keep track of the previous step. 6

7 The goal of this thesis is to cotiue to explore the idea of quatum walks with memory by expadig the memory of the particle to keep track of the previous two positios. Usig two-step memory meas that there are ow four differet possible histories that the particle ca have, ad the operator ca act differetly for each of these. We ivestigate the case of two-step memory quatum walks ad how their behavior differs from the oe-step memory ad coi quatum walks. Sectio further develops this idea ad shows the process required to solve for the probability distributio of the particle at some later time. 7

8 Methods The goal of this sectio is to describe a method for solvig the time evolutio of a quatum walk with two-step memory i a oe-dimesioal system. First we cosider the case of a walk o a ifiite positio space ad defie the evolutio operator that will be used i this work. This sectio also describes the mai steps i the process of calculatig how the state evolves, icludig the use of the spatial Fourier trasform, diagoalizatio of the operator i the Fourier space, ad fially the use of the iverse Fourier trasform. Solvig the oe-dimesioal walk results i complicated expressios that caot be solved aalytically. These expressios are solved umerically usig Mathematica ad the results are preseted i Sectio 3..1 Quatum Walk o a Ifiite Positio Space Cosider a particle that is free to move i oe-dimesio alog a ifiite space of discrete positios, Z. Betwee ay cosecutive time steps the particle ca move either to the left or the right. Movig to the left is cosidered movig i the egative directio ad movig to the right is cosidered movig i the positive directio. Sice this is a quatum system, as log as o measuremets of the particle s positio are take, the state of the particle will spread out i both possible directios. This sectio describes how to calculate the probability distributio for the positio of the particle at a later time. The major differece betwee the quatum walk described i this paper ad covetioal quatum walks is that it uses a two-step memory. Two-step memory meas that the quatum state icludes the previous two positios of the particle i additio to the curret positio. The quatum operator the acts differetly o the state depedig o the previous positios. Before we ca itroduce this operator ad how it acts o a state, we must distiguish betwee the four possible memory states. These states are more easily visualized by describig them each as a tail that poits from the two previous positios to the curret positio.. Types of Tails The term tail refers to the part of the particles state that keeps track of its previous two positios. Sice there are two possible directios for the particle to move at ay time, there are four possible paths it ca take over two time steps. For example, cosider a particle that is curretly at a positio. The particle could have moved two steps i the positive directio, arrivig at from. It could also have begu by movig i the egative directio from, chaged directio, ad the moved back to from 1. Similarly, the walk could arrive at from the positive directio i two differet ways. All four of these types of tails are defied i the diagram below. (right-right) ( ) ( 1) state RR (left-right) ( 1) state LR (right-left) ( + 1) state RL (left-left) ( + ) ( + 1) state LL Thus we may ow refer to the memory state, or tail, i terms of the four basis kets give above. I developig the evolutio operator for this system, we also be refer to the arrows of the tail. 8

9 These arrows are exactly those see i the diagram above ad will be described as either left or right poitig arrows. The first arrow i a tail is the older step that the particle took ad is listed as the first etry i the tail ket. The secod arrow i a tail is the most recet step that the particle took to arrive at its curret positio ad is give by the secod etry i the tail ket. The full state of the particle is ow give by the tesor product of the tail ad positio states of the particle. ψ = tail (.1) For example, cosider a particle i the pure state RR 0. The particle is curretly at positio = 0 ad it arrived there by takig two steps to the right. So this particle arrived to = 0 from =. Sice this is a quatum system, the particle ca also be i a superpositio of states. This ca be a superpositio of the tail states such as ψ = 1 ( RR + LL ). However, superpositios of tail states ad the positio of the particle such as ψ = 1 LL RR + 1 are also possible. The goal ow is to defie the quatum evolutio operator ad how it will act o the states..3 Evolutio Operator Give that the particle is i some state at time t, the evolutio of the state is described by ψ t+1 = U ψ t (.) where U is a uitary operator. Uitary meas that the evolutio of the system is reversible, ad all time evolutio operators i quatum mechaics must behave this way. The full evolutio operator cosists of two separate operators: the coi operator Q which operates i the tail space, ad the shift operator S which operates i the positio space. The coi operator assigs probabilities to move from oe tail to the ext based o the previous two steps. The shift operator the chages the particle s positio to match the step that was implied by the chage i the tail. First, we describe the coi operator i more detail. The behavior of the operator is most easily described by referrig to the separate arrows i the tail. First, otice that give some tail at time t, there are oly two possible tails that the particle ca have at time t + 1. This is because the secod arrow at time t will become the first arrow at time t + 1. For example, a particle with tail LR at t ca have oly tails RR or RL at t + 1. So the secod arrow i a particle s tail determies the first arrow i its tail after oe time step. The coi operator assigs the probability that the particle will move either left or right durig a time step. For this quatum walk, we defie the coi so that there is a positive probability p < 1 that betwee t ad t + 1 the particle moves i the same directio as the first arrow i the tail at t. The the probability that the particle moves the opposite directio as the first arrow i its tail is 1 p. For example, if the particle has a tail RL, the the first arrow is to the right. This meas 9

10 there is a probability p that its ext step is to the right ad a probability p 1 that its ext step is to the left. The operator ca also be thought of as the probability that the tail state maitais its recet behavior. If it cotiued movig i the same directio at the previous time step, the the probability that it keeps movig i that directio is p. If the particle chaged directio at the previous time step, the there is a probability p that it will chage directio agai i the curret step. We ca ow write how the coi operator acts o each of the four tail kets. Q RR = p RR + 1 p RL Q LR = 1 p RR + p RL Q RL = p LR + 1 p LL Q LL = 1 p LR + p LL. (.3) These four relatios ca also be writte i matrix otatio. RR t+1 RR t p 1 p 0 0 LR t+1 RL t+1 = Q LR t RL t, Q = 0 0 p 1 p 1 p p 0 0. (.4) LL t+1 LL t p p There are a few commets to be made o the operator Q. First, the terms p ad 1 p are used istead of p ad 1 p because the operator must be expressed i terms of the probability amplitudes. The actual probabilities to measure the associated tails are the squares of the probability amplitudes, which would give p ad 1 p as required. Also, egative sigs have bee used o two of the etries. This was ecessary to make the operator uitary, but it does ot affect the probabilities. Fially, otice there are oly two ozero etries i each colum of the operator Q. This reflects the fact that ay tail has oly two possible chages it ca make betwee time steps. Now that the coi operator has bee defied, we move o to the shift operator S. The behavior of the shift operator is closely related to the coi operator because the way i which the tail state chages implies the directio that the particle moved. For example, if the tail chages from RR to RL durig some time step, the the particle must have moved to the left durig that step. This meas that the shift operator must decrease the positio by oe durig that step. Similarly, if the tail implies that the particle moved to the right, the shift operator must icrease the positio by oe. The shift operator ca ow be defied as S =Π RR Π LR Π RL 1 + Π LL 1, (.5) where Π ij are the projectio operators correspodig to the tail states. The role of the projectors is to select oe of the tail states, ad the the sums shift all positios with that tail state i the appropriate directio. Cosider the first of the four terms i the shift operator. The projector Π RR selects all the parts of the state that have the tail RR. The, sice that tail implies that the most recet step was to the right, the sum shifts all the positios from to + 1. Fially, the coi ad shift operators must be combied to give the total evolutio operator U. U = S(Q I) (.6) 10

11 The term i paretheses operates first, mixig the tail states ad assigig probabilities for the particle to move either left or right. This term icludes the idetity operator because the actio of the coi does ot yet chage the positio states. Rather, the coi operator Q chages the tail states ad determies which directio the positios will be shifted. The shift operator S the acts o the positio of the particle to move it i the correct directio. So the overall evolutio of the state of the particle is a series of mixig the tail states ad the shiftig the positios. A example help to will illustrate how the total evolutio operator works. Cosider a particle that starts i the state ψ 0 = 1 RR ( ). (.7) This particle is i a equal superpositio of the positios = 1 ad = 4, ad both parts are i the tail state RR. To evolve the state, we first act with the coi ad the idetity. [ ] 1 (Q I) ψ 0 = (Q I) RR ( ) = 1 Q RR I ( ) = 1 ( p RR ) + 1 p RL ( ) p 1 p = RR ( ) + RL ( ) (.8) Now to complete the evolutio, we apply the shift operator to this state. Notice that oly the terms from the shift operator that have the Π RR ad Π RL projectors will be able to operate. ψ 1 = S(Q I) ψ 0 [ = Π RR Π RL 1 [ ] p 1 p RR ( ) + RL ( ) [ ] [ ] p 1 p = RR + 1 ( ) + RL 1 ( ) p 1 p = RR ( + 5 ) + RL ( ) 1 p p 1 p p = RL 0 + RR + RL 3 + RR 5 (.9) There are ow four differet terms i the state. The parts of the particle s state that were at = 1 ad = 4 each split, movig i both the left ad right directios. The probability that the state would move i either directio was assiged by the coi operator. From this simple example, it is already clear that the state of the particle expads rapidly ad the expressios become very complicated. Evolutios after a large umbers of time steps would be extremely difficult to calculate by simply applyig the operator U may times. Aother method utilizig the Fourier trasform is ecessary to calculate the state at much later times. ] 11

12 Now we defie the otatio that will be used to express ay possible state of the particle. Cosider a geeral state at time t give by ψ t = ( ) a t () RR + b t () LR + c t () RL + d t () LL. (.10) Z The state of the particle is i a superpositio of all the possible positios. I additio, at each positio the particle is i a superpositio of each of the four tail states. We have defied a t ()...d t () as the amplitudes of each tail state at the positio. Applyig the operator U gives the probability amplitudes at t i terms of the amplitudes at the previous time t 1. a t () = p at 1 ( 1) + 1 p b t 1 ( 1) b t () = p c t 1 ( 1) + 1 p d t 1 ( 1) c t () = 1 p a t 1 ( + 1) p bt 1 ( + 1) d t () = 1 p c t 1 ( + 1) + p dt 1 ( + 1). (.11) The goal is to solve these recurrece relatios so that they ca be applied to some iitial state to predict the amplitudes at a later time. Oce all four of the tail amplitudes are kow as a fuctio of the positio, the the probability distibutio for fidig the particle at is give by P t () = ψ t = ( ) a t () RR + b t () LR + c t () RL + d t () LL Z ( ) = a t () RR + b t () LR + c t () RL + d t () LL Z P t () = a t () + b t () + c t () + d t (). (.1) Therefore, the probability distributio for the positio of the particle is give by the sum of the squares of each of the four tail state probability amplitudes. The four recursio relatios above ca be solved to fid the amplitudes at ay time, but the solutio requires the use of the Fourier trasform. This trasforms the amplitudes from fuctios of positio,, to fuctios of spatial frequecy, k. I the Fourier space the evolutio operator ca be diagoalized, ad this allows us to aalytically solve for the amplitudes after ay umber of time steps..4 Fourier Trasform Due to the traslatioal ivariace of the positio space, the evolutio of the quatum walk has a much simpler represetatio i the Fourier space. Therefore, the geeral strategy for solvig the recurrece relatios is to first use the Fourier trasform o each of the four equatios. These equatios are the combied ito a matrix form for the evolutio operator i the Fourier space. We the diagoalize the operator so that it ca easily be raised to the power of t. This greatly simplifies the algebra because whe a diagoal matrix is raised to some power, each diagoal elemet is just 1

13 raised to that power. The diagoalized operator is applied to some iitial state to get the amplitudes at a later time. Fially, we carry out the iverse Fourier trasform to retur the amplitudes the positio space. The Fourier trasform is defied as f(k) = Z f()e i k, (.13) We start by applyig the trasform to the equatio for the amplitude a(). â t (k) = a t ()e i k Z = Z[ pa t 1 ( 1) + 1 pb t 1 ( 1)]e i k (.14) Distributig the expoet ad rearragig we ca rewrite Eq. (.14) as â t (k) = pe i k Z a t 1 ( 1)e i k ( 1) + 1 pe i k Z b t 1 ( 1)e i k ( 1). (.15) Chagig the idex from to 1 has o effect o the sum sice the sum rus from egative to positive ifiity. So the terms i the sums are themselves Fourier trasforms. The â t (k) coefficiet ca fially be expressed as â t (k) = pe i k â t 1 (k) + 1 pe i k bt 1 (k) (.16) Trasformig the other three coefficiets from (.11) i a similar way we get bt (k) = pe i k ĉ t (k) + 1 pe i k dt (k) ĉ t (k) = 1 pe i k â t (k) d t (k) = 1 pe i k ĉ t (k) + pe i k bt (k) pe i k dt (k). (.17) We ca ow see why the Fourier trasform helps with the calculatio. The four recursio relatios i the Fourier space are oly fuctios of k. However, the recursio relatios from (.11) are i terms of 1,, ad + 1. Combiig (.16) ad (.17) we get the matrix M, the evolutio operator i the Fourier space. â t bt v t = ĉ t = M d t â t 1 bt 1 ĉ t 1 d t 1 â 0 = M t b0 ĉ 0 d 0 (.18) p e i k 1 p e i k p e i k 1 p e i k M = 1 p e i k p e i k 0 0. (.19) p e i k p e i k The state vector v t has bee defied as the vector whose compoets are the four probability amplitudes â t... d t. The matrix M is ow diagoalized to make it easier to solve for v t which will give the probability amplitudes at some later time. I the ext subsectio.5 we solve for the four eigevalues ad eigevectors which are the used to form the diagoal operator. 13

14 .5 Diagoalizig the Operator M The first step i diagoalizig the matrix M is to solve for the eigevalues. The four eigevalues are obtaied by solvig the equatio det(m Iλ) = 0 (.0) for λ. Takig the determiat of p e i k λ 1 p e i k λ p e i k 1 p e i k M = 1 p e i k p e i k λ 0, (.1) p e i k p e i k λ gives the characteristic equatio λ 4 ( p e i k + p e i k) λ 3 + ( p e i k + p e i k) λ 1 = 0. (.) Usig the expoetial form of the cosie fuctio we ca rewrite Eq. (.). λ 4 ( p cos k)λ 3 + ( p cos k)λ 1 = 0. (.3) Fially, solvig this equatio for the eigevalues we get λ 1 = 1, λ = 1 λ 3 = p cos k + i 1 p cos k, λ 4 = p cos k i 1 p cos k. (.4) At this poit it is helpful to ote that the eigevalues of a uitary operator must always have a orm of oe, so λ i = 1. This meas that the two complex eigevalues must be poits o the uit circle ad the four eigevalues ca the be expressed as λ 1 = 1, λ = 1, λ 3 = e iω k, λ 4 = e iω k, where ω k = cos 1 ( p cos k). (.5) Now that the eigevalues are kow, the eigevectors V i of the matrix M ca be foud by solvig (M Iλ i ) V i = 0 p e i k λ 1 p e i k λ 1 0 q e i k 1 q e i k 1 p e i k α i p e i k λ 0 β i = q e i k q e i k λ 0 γ i (.6) Solvig Eq. (.6) for each of the coefficiets α i, β i, γ i gives the equatio for the eigevectors i terms of the eigevalues λ i. α i = λ ie ik p, β i = λ 1 i e ik pe ik, ad γ i = 1 p 1 p pe 3ik λ 1 i e ik pe ik. (.7) λ i 14

15 The expressios for α i, β i, γ i give by Eq. (.7) are elemets of the eigevectors ad become importat whe calculatig each of the tail amplitudes. The geeral expressio for the eigevectors is 1 ( 1 λi e α ik ) p i v i = β i = 1 p ( λ 1 i e ik ) pe ik. (.8) 1 p γ i ( pe ) 3ik λ 1 i e ik pe ik λ i We ow write the four eigevalues ad the correspodig eigevectors. λ 1 = 1, λ = 1, λ 3 = e iω k, λ 4 = e iω k, where ω k = cos 1 ( p cos k) (.9) v 1 = v 3 = 1 e ik p 1 p e ik pe ik 1 p e ik 1 e iω k e ik p 1 p e iω k e ik pe ik 1 p pe 3ik e iω k e ik pe ik e iω k, v =, v 4 = 1 e ik p 1 p e ik pe ik 1 p e ik 1 e iω k e ik p 1 p e iω k e ik pe ik 1 p pe 3ik e iω k e ik pe ik e iω k. (.30) Usig the eigevalues ad eigevectors, we costruct the diagoalized operator. The best way to do this without havig to chage from the v i basis is to use the spectral decompositio of M. This is give by M = 4 i=1 λ i v i v i v i v i. (.31) The spectral decompositio is a sum of the projectio operators for each of the eigevectors weighted by the eigevalues. The ier products i the deomiator are required because the four eigevectors obtaied above were ot ormalized. The real beefit of this form is that raisig it to the power t simply raises the eigevalues to that power. M t = 4 i=1 (λ i ) t v i v i v i v i. (.3) We fially have everythig we eed to calculate how the state of the particle will chage after ay umber of time steps. If v 0 is the vector that gives the iitial tail amplitudes i the Fourier space, the after t time steps the state will be v t = M t v 0 = ( 4 i=1 (λ i ) t v i v i v i v i ) v 0. (.33) 15

16 For simplicity, we assume that the particle starts out i the pure state RR. So the iitial vector will be the uit vector v 0 = e 1. Sice each of the vectors v i were chose so that their first etry was oe, this meas that v i v 0 = v i e 1 = 1. Give the iitial state RR, the fial state of the particle i the Fourier space will be 1 v t = 4 i=1 (λ i ) t v i v i v i â t bt ĉ t = d t 4 i=1 (λ i ) t v i v i α i β i γ i. (.34) Fially, this gives the four equatios for the tail amplitudes after the particle has evolved for a time t. â t = ĉ t = 4 i=1 4 i=1 (λ i ) t v i v i, bt = (λ i ) t v i v i β i, dt = 4 i=1 4 i=1 (λ i ) t v i v i α i (λ i ) t v i v i γ i (.35) Each amplitude has four terms that cotai eigevalues ad eigevectors. We have already solved for all of the elemets cotaied i these terms. However, these amplitudes are still i the Fourier space. Before we ca calculate the probability distributio for the particle, we must use the iverse Fourier trasform to retur each of the terms above to the positio space..6 Iverse Fourier Trasform The iverse Fourier trasform is defied as f() = 1 π π 0 f(k)e ik dk (.36) This iverse trasform must be applied to each of the expressios i (.35) to fid the tail amplitudes as fuctios of positio. Fially, these amplitudes will be used to calculate the probability distributio for the positio of the particle usig Eq. (.1) P t () = a t () + b t () + c t () + d t (). The probability distributios is calculated umerically ad preseted i the Sectio 3. 16

17 3 Results This sectio presets the plots for the tail amplitudes ad the total probability distributio of the particle. Each result is evaluated at time steps 99, 100, 199, ad 00. These times were chose to show evolutio over a sigle step ad over larger time itervals. Whe calculatig the evolutio of the particle, oe must first fid the four probability amplitudes a t () through d t (), so the plots for these are preseted first. The amplitude plots share similar features that also appear i the probability distributio. 3.1 Tail Probability Amplitudes I Sectio.5 we solved for each of the tail amplitudes i terms of t ad k. To retur to the origial positio space, we apply the iverse Fourier trasform defied by Eq. (.36) to each expressio from (.35). The itegral for the a t () is a t () = 1 π π = 1 π 0 π 0 [ 4 [ i=1 ] (λ i ) t dk v i v i 1 p 4(1 p cos k) + 1 p ( 1)t 4(1 + p cos k) + e +t(iω k) 1 + p p cos k + ( p si k) 1 p cos k 4 ( 1 p cos k ) + e t(iω k) 1 + p p cos k ( p si k) 1 p cos k 4 ( 1 p cos k ) ] e ik dk (3.1) where ω k ad the eigevalues λ i are give by Eq. (.5). The full itegral from Eq. (3.1) is ot solved aalytically, but it is still a exact solutio for the tail amplitude. Istead of solvig aalytically, the expressio is itegrated umerically ad Fig. 1 shows the result for the itegral. The other three probability amplitudes are calculated i the same way as a t () by applyig the iverse Fourier trasform. The itegral expressios are extremely similar to Eq. (3.1), except that each term is multiplied by α i for b t (), β i for c t (), ad γ i for d t (). b t () = 1 [ π 4 ] (λ i ) t π 0 v i=1 i v i α i dk, c t () = 1 π [ 4 ] d t () = 1 π π 0 i=1 (λ i ) t v i v i γ i π 0 [ 4 i=1 ] (λ i ) t v i v i β i dk dk (3.) The factors α i, β i, ad γ i are give by Eq. (.7). These coefficiets are also calculated by umerical itegratio, ad the results are preseted i Figs., 3, ad 4. 17

18 a t a t At tim e step : 99 At tim e step : 100 a t a t At tim e step : 199 At tim e step : 00 Figure 1: Probability amplitude a t () as a fuctio of positio for time steps 99, 100, 199, 00. b t b t At tim e step : 99 At tim e step : 100 b t b t At tim e step : 199 At tim e step : 00 Figure : Probability amplitude b t () as a fuctio of positio for time steps 99, 100, 199,

19 c t c t At tim e step : 99 At tim e step : 100 c t c t At tim e step : 199 At tim e step : 00 Figure 3: Probability amplitude c t () as a fuctio of positio for time steps 99, 100, 199, 00. d t d t At tim e step : 99 At tim e step : 100 d t d t At tim e step : 199 At tim e step : 00 Figure 4: Probability amplitude d t () as a fuctio of positio for time steps 99, 100, 199,

20 3. Probability Distributio The four tail amplitudes have bee calculated, so we ca fially use Eq. (.1) to calculate the probability distributio of the particle as a fuctio of time ad positio. Fig. 5 shows the results for the probability. P t () = a t () + b t () + c t () + d t (). 0.4 p robability Tim e step : probability Tim e step : site site 0.4 p robability Tim e step : probability Tim e step : site site Figure 5: Probability distributio P t () for the particle as a fuctio of positio for time steps 99, 100, 199, 00. 0

21 4 Discussio We ow discuss the importat features of the results preseted i Sectio 3. We start by discussig the probability amplitudes for the tail states, ad the cosider the overall probability. The probability distributio for the DTQW with two-step memory shows behavior typical of quatum walks, but we also see the emergece of classical behavior. 4.1 Tail Probability Amplitudes Calculatig the tail probability amplitudes was the focus of most of the work i the Sectio because we must kow each of them before calculatig the probability. However, they are ot as physically meaigful as the probability distributio because they do ot actually represet the probability to fid the particle at a give positio ad time. Istead, they give oly that probability that a give tail state is measured. For example, if we had some way of measurig the curret positio of the particle as well as the previous two positios, the a t () would allow us to calculate the probability that we measure the RR tail state. I additio, the plots i Figs. 1,, 3, ad 4 are probability amplitudes, ot probabilities. This is why the amplitude values are able to take o both positive ad egative values. The tail amplitudes do show the same features that are see i the probability distributio. First, Figs. 1,, 3, ad 4 all show a sharp cetral peak located aroud the origial locatio of the particle. This implies that there is a strog probability that the particle would be foud at the same positio it started at. The cetral peak does oscillate betwee time steps, but it does ot after large amouts of time. Mathematically, the statioary peak is caused by the first two terms i each of the itegrals. For example, the a t () coefficiet is give by Eq. (3.1) where we see that the first two terms are oly time depedet due to the ( 1) t. The itegral of these two terms will be differet for positive or egative t, but it does ot deped o how much time has passed. The 1 ad -1 eigevalues lead to similar statioary cetral peaks i each of the other three tail amplitudes as well. The results for the tails also show peaks i the amplitude that move away from the origial locatio of the particle. Fig. 1 illustrates this clearly with a peak propagatig to the right with time. We ca also see that the peak propagates ballistically, or liearly with t. At time step t = 100 the peak appears aroud = 70, while at time step t = 00 the peak is aroud = 140. Also, the amplitude of the propagatig peaks is differet for each of the four plots. For example, the propagatig peak i a t () is always larger tha i b t () or c t () (Figs. ad 3), ad d t () (Fig. 4) has such small peaks that we do ot eve the directioality that we did i the other plots. This behavior is see because of the iitial state of the particle ad shows that the iitial state has a strog effect o the evolutio of the particle. Fially, otice that for each of the figures, half of the poits are always zero. This must be the case because the particle ca oly chage its positio by oe at ay time step. For example, if the particle starts at a eve positio, the it must move oe step to the left or right at the ext time step ad must ed o a odd positio. No matter how may steps are take, the particle ca oly be foud at a eve or a odd umbered positio. This still applies to our quatum particle, eve though it spreads out over may positios as it evolves. 1

22 4. Probability Distributio The probability distributio Fig. 5 is the fial result for the quatum walk with two-step memory, ad gives the actual probability that the particle will be measured at a give time ad positio. It is calculated from the tail amplitudes accordig to Eq. (.1) ad shows the same major features that were see i the probability amplitudes: ballistic propagatio ad a statioary cetral peak. The peak that propagates to the right spreads ballistically, ad this is a very commo feature of other quatum walks. However, the propagatio of a DTQW with two-step memory is much more directioal tha most other walks. Fig. 5 shows that there is a extremely small probability that the particle is foud to the left of its origial positio, ad the directio i which the particle is likely to propagate depeds o the iitial tail state of the particle. For example, if we had chose the iitial state to be LL, the the propagatio of the particle would be predomiately to the left. The ability to direct the propagatio could be very useful whe cosiderig the algorithmic uses of the quatum walk with two-step memory. The other major feature i the probability distributio is the statioary peak aroud the origial positio of the particle. This is ot a feature that is commoly see i quatum walks ad represets the emergece of classical behavior. This meas that keepig track of the added step of memory is likely the begiig of a trasitio to a classical system. Oe explaatio is that whe keepig track of oe more positio, we must kow the exact path that the particle took over the last two time steps. This elimiates some of the possible paths i the path itegral, thus decreasig the iterferece betwee paths. This is a very iterestig result ad should be explored further. Similar statioary peaks have bee see i the paper by Bru, Carteret, ad Ambaiis []. They also suggest that they are trasitioig to classical behavior, but they do so i a much differet way tha preseted above. They use multiple quatum cois to decrease the iterferece ad fid that oly i the limit of a differet coi at each time step does the quatum behavior of the particle disappear.

23 5 Coclusio The framework for quatum walks with memory has prove useful for solvig some problems that have ot yet bee solved usig stadard coi methods. This thesis explored the effect of extedig the memory of the quatum state. The walk preseted is a discrete-time quatum walk with twostep memory, meaig that the state of the particle keeps track of its curret positio as well as its previous two positios. The walk defied propagates o a oe-dimesioal discrete positio space, so the effect of this exteded memory is that there are four possible paths by which a particle ca reach a give positio. We defie these as the four possible tail states ad they are aalogous to the two coi states commoly used whe solvig quatum walks. The evolutio operator for the walk describes how the particle propagates i oe directio or the other depedig o the tail state. This walk is the solved to give the probability distributio for the particle as a fuctio of time step ad positio. The walk preseted above is defied o a ifiite positio space with a specific iitial state ad a geeral method is preseted for solvig such a walk. The method for calculatig the probability distributio of the particle focuses maily o solvig for the probability amplitudes of each possible tail state. First oe must use the evolutio operator to geerate four recursio relatios for the tail amplitudes. These relatios express each amplitude i terms of the tail amplitudes at the previous time step. To solve the recursio relatios, oe eeds to use the Fourier trasform to express the equatios i the Fourier space ad the express the four equatios i matrix form. It is the possible to diagoalize the matrix ad calculate the probability amplitudes after ay umber of time steps. Fially, the iverse Fourier trasform is applied to express each of the amplitudes i the origial positio space ad the probability amplitudes ca be combied to yield the probability distributio for the particle. The four tail amplitudes are very complicated itegrals that we do ot solve aalytically. Istead, the itegrals were evaluated umerically at differet time steps to geerate the plots for the amplitudes as a fuctio of positio. These plots are the combied to calculate the probability distributio for the particle. Although most of the work goes ito solvig for the four tail amplitudes, these plots are less physically meaigful tha the probability distributio which actually gives the likelihood that the particle is foud at a give time ad positio. The probability distributio for the particle shows two mai features. First we have ballistically propagatig peaks which are a commo feature of quatum walks. These peaks spread at a rate directly proportioal to the time, ulike the diffusive propagatio typical of classical walks i which the rate is proportioal to t. However, the propagatio i the case of the two-step memory walk appears to be much more directioal tha see i other quatum walks. This directioality obviously depeds o the iitial state of the particle, so oe could chage the directioality by choosig a iitial state with a tail that poits i a give directio. This behavior could prove to be useful whe cosiderig the algorithmic uses of quatum walks with memory. The other mai feature of the probability distributio is a statioary peak located at the iitial positio of the particle, ad this is ot a commo feature see i quatum walks. Mathematically, the cetral peak is caused by the eigevalues 1 ad -1 that were foud whe diagoalizig the evolutio operator. The statioary states associated with these eigevalues do ot decreases i amplitude as the time step icreases. Istead, they create a sharp arrow peak that oscillates betwee successive steps but does ot dimiish with time. Physically, the cetral peak represets 3

24 the emergece of a classical feature i this quatum walk ad suggests that keepig track of two steps of memory may be the start of a trasitio to a classical system. Oe explaatio for this is that keepig track of the previous positios reduces some of the iterferece i the path itegral for the particle because the path it took over the previous two steps is kow. Similar statioary peaks i the probability distributio have bee see i the work of Bru, Carteret, ad Ambaiis, ad they also claim that this is a trasitio to classical behavior []. However, their approach is much differet tha the oe preseted above. Istead they use multiple differet coi operators ad coi operators of higher dimesio to dimiish the iterferece i the path itegral. Future work will cotiue to explore the properties of quatum walks with memory. First, it would be iterestig to cosider how the particle would evolve if allowed to propagate o a fiite oe dimesioal space, such as a rig. I the case of periodic boudary coditios, the propagatig peaks i the probability distributio will move aroud the rig istead of spreadig out ifiitely. This would allow the peaks to iterfere with oe aother, ad it is ot clear what features to expect i the fial probability distributio. We would also like to explore the effects of extedig the memory of the particle to a larger umber of steps. The quatum walk with two-step memory showed classical behavior, ad it is importat to ivestigate whether this trasitio to a classical radom walk will cotiue as we icrease the memory to keep track of a large umber of steps. 4

25 6 Ackowledgemets I would like to thak Professor Yevgeiy Kovchegov ad Zlatko Dimcovic for givig me the opportuity to work o such a iterestig project. Their framework for DTQWs was the motivatio for the ivestigatio of quatum walks with two-step memory. Yevgeiy s kowledge ad isight guided every step of the work above ad Zlatko provided oe-o-oe help o everythig from uderstadig the Hadamard walk to applyig the Fourier trasforms. Zlatko s Mathematica program was also used to geerate the plots preseted i the Results sectio, ad his istructio o usig LaTex was ivaluable while writig this thesis. I would also like to thak Professor Jaet Tate for her advice ad feedback o writig this thesis. Her suggestios o style, orgaizatio, ad cotet throughout the year have greatly improved my skills as a writer. Fially, I would like to thak the Research Office at Orego State Uiversity for gratig me a full year Udergraduate Research, Iovatio, Scholarship ad Creativity (URISC) award. The URISC fudig provided fiacial support which allowed me to work o this project throughout the etire school year. 5

26 7 Refereces [1] A. M. Childs, O the relatioship betwee cotiuous- ad discrete-time quatum walk, Commu. Math. Phys. 94, 581 (009). [] Todd A. Bru, Hilary A. Carteret, ad Adris Ambaiis, Quatum walks drive by may cois, Phys. Rev. A 67, (003). [3] J. Kempe, Quatum radom walks: A itroductory overview, Cotemp. Phys. 44, 307 (003). [4] A. Nayak ad A. Vishwaath, e-prit quat-ph/ [5] Daiel R. Simo. O the power of quatum computatio. SIAM Joural o Computig, 6(5): , October [6] Peter W. Shor. Polyomial-time algorithms for prime factorizatio ad discrete logarithms o a quatum computer. SIAM Joural o Computig, 6(5): , October [7] Lov K. Grover. A fast quatum mechaical algorithm for database search. I Proceedigs of the Twety-Eighth Aual ACM Symposium o the Theory of Computig, 1-19, (1996). [8] A.M Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutma, ad D.A Spielma, Expoetial algorithmic speedup by quatum walk, Proc. 35th ACM Symposium o Theory of Computig, pp , (003). [9] Z. Dimcovic, D. Rockwell, I. Milliga, R. M. Burto, T. Nguye, ad Y. Kovchegov, Framework for discrete-time quatum walks ad a symmetric walk o a biary tree, Phys. Rev. A 84, (011). 6