An analytic solution for one-dimensional quantum walks

Transcription

1 An nlyic soluion for one-dimensionl qunum wlks In Fuss, Lng Whie, Peer Shermn nd Snjeev Nguleswrn. School of Elecricl nd Elecronic Engineering, Universiy of Adelide, Ausrli. Deprmen of Aerospce Engineering, Iow Universiy, Unied Ses of Americ The firs generl nlyic soluions for he one-dimensionl wlk in posiion nd momenum spce re derived. These soluions revel, mong oher hings, new symmery feures of qunum wlk probbiliy densiies nd furher insigh ino he behviour of heir momens. The nlyic epressions for he qunum wlk probbiliy disribuions provide mens of modelling qunum phenomen h is nlogous o h provided by rndom wlks in he clssicl domin.. Inroducion.. Bckground There is growing body of evidence h qunum wlks hve similr role in he modelling of qunum phenomen nd he developmen of lgorihms for qunum compuion s h of rndom wlks in he clssicl domin [-5]. This evidence hs been ccumuled in pr hrough numericl, momen nd limi nlysis of he discree one-dimensionl qunum wlk. The eisence of generl nlyic soluion for he discree qunum wlk hs been n open quesion hus fr. In his pper, such soluion for he posiion nd momenum spce wvefuncions of he discree one-dimensionl qunum wlk is derived. In conrs o previous nlyses of qunum wlks h hve lso uilised he momenum spce our nlysis provides elemenry closed form soluions for boh he momenum nd posiion spce. These soluions re powerful ools for nlysing he properies of qunum wlks s well s providing cpbiliy for modelling qunum phenomen. Qunum wlk models hve n ineresing nd imporn pplicion wih regrd o wheher humn decision nd recion ime d mnifes qunum effecs [6, 7]. In his cone he lineriy of he qunum wlk probbiliy disribuions, derived in his pper, wih respec o heir symmery prmeers should enble hem o be esimed from eperimenl d vi simple echniques, while he isolion of he emporl evoluion prmeer enbles is esimion vi one-dimensionl serch. Previous work h hs been moived by he developmen of lgorihms for qunum compuing hs sough insigh for his by choosing prdigmic emples of qunum wlks, such s he Hdmrd wlk, nd rgued h hese re ypicl of qunum wlks s whole [, 3]. The work of his pper in developing simple eplici nlyic forms of he generl wve funcions nd hence probbiliy disribuions, h re

2 closely conneced o he underlying prmeers, conrss o his prdigmic pproch. The resuls obined rise quesions bou how ypicl he Hdmrd wlk is of he specrum of discree qunum wlks. Hence while i ws no he min moivion in seeking hese generl soluions i is possible h hey will provide beer bsis for developing qunum lgorihms hn he pproches used hus fr. For emple he isolion of he emporl evoluion prmeer in he wlk from he oher wo symmery prmeers my ssis in developing opimision sregies for lice serches o loce priculr vlues of prmeer [9] or o deermine sisfcory pln []... Srucure In he ne secion he qunum wlk dynmic equions in posiion spce re sed nd used o obin hose for he momenum spce. Then in secion 3 generlised de Moivre principl is used o derive he momenum-spce ime-evoluion operor in erms of ype II Chebyshev polynomils. Anlyic epressions for he momenum spce φ ( p,) nd posiion spce ψ (,) qunum wlk wvefuncions re obined using his ime-evoluion operor in secion 4. The posiion spce wvefuncions re lef in prilly bsrc form in his secion o ssis in providing generl nlysis of he symmery properies of he qunum wlk posiion spce probbiliy densiies ρ (,) in secion 5. The key resul of secion 5 is h he qunum wlk probbiliy disribuions hve he specil form of sum of even nd odd funcions. These odd funcions re muliplied by he wo qunum wlk prmeers, previously referred o s symmery prmeers. An eplici nlyic represenion of he posiion spce wve funcions is presened in secion 6. This mkes use of new ype of one-dimensionl funcion whose properies re prilly nlysed o suppor he subsequen qunum wlk nlysis. In secion 7 hese funcions re used o derive nlyic epressions for he momens of he wlks. In secion 8 he resuls of he momen nlysis re used o nlyse he wlks nd provide linkges o oher resuls in he lierure. The pper finishes wih summry nd he menion of n open re of reserch.. Qunum Wlk Dynmic Equions c For given ψ (, ) δ, c = we consider he evoluion of qunum se ψ (, ) C for discree imes on line Z. [, ] The dynmics of he se evolve ccording o he difference equions ψ ik ( ) = e [ ψ (, ) + bψ (, ) ], [ ] ik * * ( ) = e b ψ ( +, ) + ψ ( +, ) ψ (), where + b =, k R nd ψ nd ψ re he componens of he spinor ψ.

3 We noe h hese difference equions define liner sysem nd hence follow Nyk nd Vishwnh in eploiing he spil homogeneiy of he qunum wlks by pplying he Fourier rnsform [] o equion o give φ φ ip ( p, ) = ψ (, ) e = [ ] ik ip ip ( p ) = e e φ ( p, ) + be φ ( p, ), [ ] ik * ip * ip ( p ) = e b e φ ( p, ) + e φ ( p, ) φ. (3), We noe h wih choice of unis so h = nd n pproprie normlision procedure such s requiring periodic boundry condiions hen he spinor () φ( p, ) φ( p, ) = φ( p, ) (4) cn be inerpreed s momenum spce wve-funcion for he qunum wlk wih p ( π, π ][]. Using his wve funcion equion 3 cn be rewrien s he mri equion where is unimodulr mri. ik ( p, ) = e S( p) φ( p, ) φ (5) S ( ) ip ip e be p = (6) * ip * ip b e e By pplying he Fourier rnsform o he iniil condiion (,) equivlen iniil condiion in he momenum spce ( p,) ( p,) (,) (, ) φ ψ = φ ψ ψ we cn obin he. (7) We cn obin he momenum spce wvefuncion n rbirry ime wih his iniil condiion by iering equion 5 o give ik ( p, ) e S ( p) φ( p,) φ =. (8)

4 This equion permis he inerpreion of i( ) k ( ( ) = e S ) ( p) T (9) s he evoluion operor in he momenum spce. This by mri polynomil provides simpler nlyic bse hn is equivlen he infinie dimensionl consn mri in he posiion spce. 3. The Evoluion Operor Oher uhors hve nlysed he momenum spce wve funcions using n eigenvlue decomposiion of his evoluion operor [, 3]. We use n lerne pproch h hs he dvnge of giving he generl evoluion operor in simple nlyic form nd hence generl elemenry closed forms for he momenum nd posiion spce wvefuncions. The unimodulr mri S( p) cn be wrien in n eponenil form s S(p) = Ep(iθ(p)c(p).σ) = cos(θ(p))i + isin(θ(p))c(p).σ () where θ nd c re rel funcions of p nd he mri vecor σ hs Puli mri componens [3, 4] σ = i, σ = nd () σ 3 = i nd I is he ideniy mri. This eponenil form llows us o cn wrie he powers of S( p) s S ( p) = Ep(iθ(p)c(p).σ) = cos(θ(p))i + isin(θ(p))c(p).σ. () The rigonomeric epressions in his equion cn be wrien in erms of he Chebyshev polynomils T n nd U n s [5] nd ( θ ) ( cos( θ )) cos = T ( θ ) U ( cos( θ )) sin( θ ) sin =. (3) Subsiuing hese epressions ino equion gives S ( p) T ( ( θ ( p) )) I + U ( cos( θ ( p) )) i sin( θ ( p) ) = cos c(p).σ. (4) I is ineresing o noe h his form of he ime evoluion operor implies h he momenum spce represenion of he Hmilonin for he qunum wlk is H = θ(p)c(p).σ.

5 This equion cn be wrien in erms of he ype II Chebyshev polynomils lone by using he relion Tn ( ) = U n( ) U n( ) (5) o give S ( p) U ( ( θ ( p) )) I U ( cos( θ ( p) ) = cos )[ cos(-θ(p))i + isin(-θ(p))c(p).σ] (6) where we hve used he even propery of he cosine funcion nd he odd propery of he sine funcion. By compring he epression wihin he squre brckes wih equion we see i is equivlen o n eponeniion of he inverse of he one sep ime evoluion mri S ( p)nd hus wrie S ( p) = U ( cos( ( p) )) I U ( ( θ ( p) )) S ( p) θ cos. (7) By subsiuing his epression ino equion 9 we obin he ime evoluion operor s [ ] ik (,) = e U ( cos( θ ( p) )) I U ( ( θ ( p) )) S ( p) T cos. (8) We cn deermine he funcion ( θ ( p) ) defining he inner produc ( ) cos in erms of he componens of S p by, = ) (9) ( A B) Tr(AB on he vecor spce of wo by wo uniry mrices nd hence obin n inner produc spce wih { I σ, σ, } s n orho-norml bsis., σ 3 We noe h he coefficien of I on he righ hd side of equion is ( θ ( p )) he coefficien of I on he lef hnd side is ( I S( p) ) = cos( p d ), () where is he bsolue vlue of nd d is rgumen, h is cos nd id = e. () Thus using he equliy of coefficiens nd equion we cn wrie he ime evoluion operor in equion 8 s T { ( ) I U ( ( p d )) S ( p) } (,) e U cos( p d ) ik = cos. ()

6 4. Wve Funcions 4.. Momenum Spce The momenum spce wve funcions re now esily obined from equions 8, 9 nd s or using s φ φ { ( ) I U ( cos( p d )) S ( p) } φ(,) ik ( p, ) e U cos( p d ) = (3) φ ik ( p ) e U cos( p d ) p ( p, ) = S ( p) φ( p,), (4) ik ( ) φ( p,) e U ( cos( p d )) φ(, ), = p We cn wrie his more eplicily in erms of he spinor componens s i( pd ) i( pd ) [ ( ) c U ( ( p d ))( e c e )] ik ( p ) e U cos( p d ) cos. (5) φ, = β c (6) φ * i( pd ) i( pd ) [ ( ) c U ( ( p d ))( e c e )] ik ( p ) e U cos( p d ) = cos β +, c where he ler erms re derived from he epnsion of equion 4 s (6b) nd defining φ * e c ( p, ) = * ip ip b e e c ip β = be id be ip (7). (8) 4.. Posiion Spce The inverse Fourier rnsform of equion 6 gives he posiion spce wve funcions i( d + k ) ( ) e [ c ( u ( : ) u ( : + ) ) + βc u ( : ) ] ψ (9), = i( d + k ) (, ) e [ c ( u ( : ) u ( : ) ) β * c u ( : ) β c ] ψ + = * (3) where hve defined π ip u ( : ) = U ( cos( p) ) e dp (3) π π s he inverse Fourier rnsform of he ugmened Type II Chebyshev polynomils.

7 We will show in secion 6 h he funcions ( ) u cn be epressed in simple : closed form. A presen we noe h hey re rel even funcions s hey re he inverse Fourier rnsform of rel even funcions. An inspecion of equions 9 nd 3 leds us o define he funcion ( : ) = u ( : ) u ( : + ) f (3) nd hence wrie he posiion spce wvefuncions s i( d + k ) ( ) e [ c f ( : ) + βc u ( : ) ] ψ (33), = i( d + k ) ( ) e [ c f ( : ) β * c u ( : ) ] ψ. (33b), = The epressions in equions 9, 3 nd 33 show clerly h he deiled emporl dynmics of he qunum wlk depend dominnly on he single rel prmeer vi he funcions ( ) u. : 5. Probbiliy Densiies nd Momens 5.. Probbiliy Densiies The qunum wlk probbiliy densiies for ech of he spinor componens cn be wrien by using equion 33 s (, ) = c f ( : ) + bcc cos( δ ) f ( : ) u ( : ) + b c u ( : ) ρ (34) (, ) = f ( : ) c bcc cos( δ ) f ( : ) u ( : ) + b c u ( : ) ρ (34b) where ( ) rg( b) + rg( c ) rg( c ) δ = rg. (35) Thus i cn be seen eplicily h he one dimensionl qunum wlk is n inerferomeric sysem wih only hree effecive rel prmeers h cn be chosen s, c ndδ. We noe h b = nd c = c.

8 By epnding he funcions f nd f we re ble o wrie he qunum wlk probbiliy disribuions s he sum of n even [ ] : [ + ], (36) ρ even(, ) = u ( : ) + u ( : ) + u ( : + ) u ( ) u ( : ) + u ( : ) nd odd componen, α, ν ρ odd ν [ ] + (36b) where nd (, ) = u ( : ) u ( : + ) ( ν α ) u ( : ) { u ( : ) u ( : ) } + ( c c ) ν = c = (37) ( δ ) α = bc c cos. (38) We noe h he sum of even nd odd funcions is specil form for n symmeric funcion. Emples of he even nd odd componens re ploed in figures nd. Furher we noe h he qunum wlk probbiliy disribuions re liner in he prmeers ν nd α nd h hese ffec only heir symmery. These prmeers nd hence he odd componen of he disribuion re zero when nd ( ) c = (39) cos δ =. (4) Thus hese re sufficien condiions for qunum wlk o be even. 5.. Momens Since he qunum wlk probbiliy densiies cn be decomposed ecly ino he sum of n even nd odd componen hen he even momens of he qunum wlk = n (, ) = ρ ( n n = ρ, ), where n N, (4) = depend only on he even componen of he probbiliy disribuion, equion 36 nd hence only on he prmeer. Similrly he odd momens of he qunum wlk even n+, α, ν n+ ν = odd = = n+, α, (, ) = ρ ( ρ, ), where n N (4)

9 depend only on he odd componen of he probbiliy disribuion equion 36b. The form of he odd probbiliy disribuion leds nurlly o he furher decomposiion ρ = ~ ~. (43 ) (, ) ( ν α ) ρ (, ) νρ ( ), α, ν odd mi sq, wih componens ~ (, ) u ( : ) u ( : ) ρ (44) sq = + { } nd ~ (, ) u ( : ) Uˆ ( : ) u ( : ) ρ. (45) mi Thus we cn wrie he odd momens s = + n+ n+ ( ) ~ ν α ρmi (, ) ν ~ ρsq ( ) n+, α, ν =, = = 5.3. Clculions The even probbiliy disribuion equion 36 cn be simplified o ( ) = [ u ( : ) + u ( : + ) ] u ( : ) u ( even, :. (46) ρ ). (47) by using he recursion u ( ) u ( : + ) + u ( : ) u ( ) + : : which is derived in ppendi A. = (48) In order o provide comprison wih oher elemens of he lierure equions 47 nd 48 were used o clcule he probbiliy disribuion in figure. [, 3 nd 4] = α = ν = Figure. The even Hdmrd wlk probbiliy disribuion =

10 The odd componen of he probbiliy disribuion for α = is ρ ( ~ ~ ) (, ) ν ρ (, ) ρ ( ), α, ν odd mi sq, =, (49) where equion 43 ws used. This is ploed for he Hdmrd wlk = nd ν = / in Figure. = α = ν =.5 Figure. An odd componen of he Hdmrd wlk Probbiliy disribuion = By vrying eiher α or ν he symmery of he qunum wlk probbiliy disribuion is chnged by miing differing mouns of he ni-symmeric componen wih he symmeric. Figure 3 shows his effec for ν. A similr bu less significn effec resuls from vrying α. ν Figure 3. Chnge in he symmery of he Hdmrd wlk wih ν =.

11 6. Foundion Funcions An nlysis of he foundion funcions ( ) u helps develop ools o nlyse : qunum wlks such s lgebric epressions for he momens in erms of he qunum wlk prmeers. I lso helps develop insigh ino he emporl behviour of qunum wlks. We sr by considering he foundion funcions role in building up he vrious disribuions ssocied wih he qunum wlks. Figure 4 below shows he funcion u 99 :. Figure 4. The Foundion Funcion u 99 : This funcion provides he mjor elemens of he even componen of he Hdmrd wlk ime =, h ws illusred in figure. The dominn feures of his componen cn be seen in he squre of his funcion h is presened in figure 5.

12 Figure 5. The squre u 99 : The even componen of he qunum wlk probbiliy disribuion is ( ) = [ u ( : ) + u ( : + ) ] u ( : ) u ( even, : ρ ). (47) The firs componen of his epression is n offse nd ddiion of he squre of he foundion funcion -. Such n operion is well known in numericl nlysis nd signl processing o hve he effec of smoohing he funcion opered on. This effec cn be seen in figure 6. Figure 6. The smoohed squre u 99 :

13 While he resemblnce beween he smoohed squre of he foundion funcion nd he even componen of he qunum wlk illusred in figure is sriking we noe h he even componen is smooher ner he origin. This er smoohing is he resul of dding he finl erm in equion Polynomil Epressions for he Foundion Funcions We now proceed o develop polynomil epression for hese foundion funcions by noing h Type II Chebyshev Polynomils cn be s epressed s he power series [5] wih [ ] indicing he firs ineger below [ ] ( ) = m U y ( y) (5) m= m nd = m m m m ( ) (5) l where is he binomil coefficien. k ip ip By subsiuing y = ( e + e ) (5) in equion 5 nd epnding he powers using he binomil heorem we obin he epression [ ] ( ( + )) ( ( )) = m m m ip m k U cos p e. (53) m= m k= k We would like o pu his in he form wih inverse Fourier rnsform i( j )p ( cos( p) ) P ( ) e U = j (54) j= ( : ) P j ( ), j u j= where he P j ( ) re polynomils in. = δ (55)

14 Hence we chnge vribles in he second summion o j = k + m nd use he l properies = if k < or k > l o remove he m dependence from he sum nd k hence obin [ ] ( ) ( ( )) = m m ip j U cos p e. (56) m= m j= j m Reordering he summions gives [ ] m m ip( j ) U ( cos( p) ) = e (57) j= m= m j m nd hen compring his wih equion 54 obin P j ( ) [ ] m m = m= m. (58) j m We noe he following properies of he polynomils: They obey he recursion P This is derived in ppendi A. ( ) P ( ) + P ( ) P ( ) =. (59) + k k + k k The even symmery of U ˆ ( ) wih respec o implies : P k ( ) P ( ) =. (6) k For j = he summion in equion (56) runces m = nd hus ( ) P =, (6) nd for j = he summion in equion (49) runces m = nd ( ) = ( ) P (6) where we hve used =. The formuls in equions 6 o 6 give us he polynomils up o = 3.

15 For = 4hey re ugmened by for = 5 nd for = 6 4 ( ) = 3 P (63) ( ) 5 P = 3 (64) ( ) = 3 + P (65) ( ) 5 P = +. (66) 6 6 Finlly when = hen Uˆ ( : ) is he inverse Fourier rnsform of Chebyshev polynomil hence nd zero oherwise. j ( ) = P if j (67) This epression is useful for checking he generl epressions for he polynomils nd quniies derived from hem such s he momens of he qunum wlk probbiliy disribuion funcions. 7. Momens Anlyic epressions for he momens of one-dimensionl qunum wlks re clculed in he following secion by using he foundion polynomils. This llows direc comprison wih he momens of eperimenl d nd lso connecs wih he lierure on qunum wlks h is momen bsed [ nd ]. 7. Even Momens In his subsecion epressions for he normlision nd second momen of he qunum wlk re clculed using equion Normlision The normlision requiremen is = (, ) = ( = ρ ρ, ). (68) = even

16 By using equion 47 nd mking he pproprie chnges in summion vribles his cn be seen o be equivlen o u ( : ) u ( : ) u( : ) = = = (69) which leds vi equion 55 o he polynomil relion j= [ P j( )] P j( ) P j( = + ). (7) This equion cn be seen o be rivilly rue in he cse = by using equion Second Momen j= The equion = even ( = ρ, ) (7) for he second momen cn be rnsformed by using equion 47 ino = ( : ) u ( : ) u( : ) u( = u : ). (7) + = This equion cn be used wih he relion beween he foundion funcion nd polynomils given in equion 55 o clcule nlyic epressions for he second momens such s = = (73) = 4 (74) 3 = (75) = (76) nd = (77) 6 = (78) 4

17 In he specil cse of = by using equion 67 i cn be shown h =. (79) Nyk nd Vishwnh [] hve shown h in he long ime limi he second momen of he Hdmrd wlk, wih =, lso increses qudriclly wih ime. (8) Hence in he ligh of his nd equion 79 we invesige he emporl dependence of he normlised second momens In figure 4 ( M ( ) =. (8) M ) is ploed for {,,3,4,5,6 } wih odd imes s dshed lines nd even s full lines. Even for hese smll imes he normlised second momens pper o be converging owrds limi wih monoonic increse of he slope of pproch owrds he fied poin = s funcion of ime. We furher noe h he vlue of he odd ime momens = remins fied nd hus he normlised momens M odd ()( ) = converge o zero in he long ime limi. odd () M ( ) = = = 3 = 5 = 6 Figure 4. Normlised Second Momens of he Qunum Wlk We noe h his epression provides useful check of he more generl momen epressions.

18 In concluding his secion we noe wih foresigh h he oscillion in ime beween zero nd one of he second momen = is consisen wih n oscillion of he firs momen his poin. 7. Odd Momens By subsiuing from equion 55 ino he epression for he odd momens given in equion 46 we obin n epression for he odd momens in erms of he foundion polynomils n+, α, ν n+ ( 4 ν α ) P ( ) P ( )( )( j) = j= j j+ n+ [ P ( j+ )( )] ( j) ν. j= (8) 7.. Firs Momen Hence he firs momen is, α, ν ( 4 ν α ) P j ( ) P ( j+ )( )( j) ν [ P ( j+ )( )] ( j = j= j= ). (83) A he ime = he firs momen is, α, ν [ 4 ] α = ν (84) h cn be wrien in erms of he prmeers c nd δ s, c, δ ( c )( ) 4 c c cos( δ ) =, (85) by using equion 84, 37 nd 38. We noe he symmery beween nd h he momen is qudric in boh of hese prmeers. The firs momens from = o = 5 re given below,, α ν, α, ν 4 3 [ 8 4 ] α 4 c in his epression nd h if cos ( ) = = ν (88) [ ] [ + ] = ν α 4 3 4, α, ν [ ] [ ] = ν α + δ (89) (9)

19 8 6 4 [ ] [ ],, α ν = ν 5 α (9) We cn check he epressions obined from hese formuls by using he specil cse = nd α = for which h we hve obined using equion 67.,, ν = ν (9) These firs 5 momens re ploed for ν = nd α = in figure 5. The ime vlues of he grphs cn be esily deermined by noing for ν = he ermining,, vlue = corresponds o he ime. Figure 5.The firs momens erly imes These momens coupled wih he vrince given in he ne secion re very reveling of he chrcer of he qunum wlk h corresponds o priculr vlue of. For smll we noe h hese wlks cn be considered o oscille wih ime beween nd -. The verciy of his semen is born winess o by he smll vlue of he vrince for smll see figure 6. These oscillions become more dmped wih ime s is incresed up o vlue of bou.. For lrge he wlks increse linerly wih ime. The inermedie region is chrcerised by he lrges vrinces s seen in figure 6.

20 7.3 Vrince The vrince (Vr) of he qunum wlk cn be evlued from he epression [6] (, ν, α, ) = ( ) Vr. (88) The normlised vrinces Vr (, ν, α, ) / re ploed for he imes up o = 5 nd for ν = nd α = in figure 6. In inerpreing hese vrinces we noe h he qunum wlks re lmos unimodl for hese prmeer vlues., ν, α = = = 3 = 5 = 4 Figure 6. Normlised vrinces for ν = nd α = The dominn feures of he vrinces re heir zero vlues he wo eremes = nd =, nd h heir nonzero vlues beween he end poins vry coninuously wih he prmeer nd pek for lrge vlues wih rpid decrese o zero =. 8. The specrum of qunum wlks An eminion of he firs nd second momens nd he vrince indices h he qunum wlks hve five regions of differen behviour reled o he criicl poins eremes =, = nd =. In he locliy of he eremes = nd = he vrince is very smll nd he wlk is ypified by is dominn componen. For smll he wlks iniilly oscille wih ime in region bounded by = nd

21 = - wih he size of he oscillions decresing wih ime if is nonzero. As increses owrds = he bounding breks erlier imes nd fer his vlue he oscillions occur round drif o lrger vlues of. The wlks wih do no hve ny oscillions nd re incresingly domined by drif o lrger vlues of wih ime h chieves is mimum =. Thus Hdmrd wlks cn be considered s ypicl of qunum wlks wih vlues of bu no for vlues smller hn his. 9. Conclusion A generl nlysis of one dimensionl qunum wlks hs been provided h gives new insighs ino heir behviour nd ools for he modelling of eperimenl d. In priculr i hs been shown h i is possible o epress he generl one dimensionl qunum wlk, in boh he posiion nd momenum spce, in erms of simple nlyic epressions. These epressions direcly show h qunum wlks depend on hree effecive rel prmeers, wo h deermine he symmery of he wlks nd he hird h conrols he emporl evoluion. The wo h ffec symmery pper linerly in he epressions for he probbiliy densiy of hese wlks mking hem esy o esime from eperimenl d. A new ype of funcion ws presened nd prilly nlysed in erms of recursions nd power series. This funcion ws used o obin he momens of he qunum wlk s lgebric epressions of he wlk prmeers. These nlyic epressions permied n nlysis of he wlks h showed h he generl second momens converge rpidly wih ime o limiing form. They were hen used o discuss he limied sense in which he Hdmrd wlk cn be considered o be ypicl of he generl onedimensionl qunum wlk. In he ligh of he srong curren ineres in qunum compuing his nlysis rises he open quesion of wh insighs for qunum lgorihm developmen cn be glened from generl nlyic models.

22 References. A. Nyk nd A. Vishwnh Qunum Wlk on he Line, qunph/7. N. Konno,, A new ype of limi heorems for he one-dimensionl qunum rndom wlk,qun-ph/63 3. T. A. Brun, H. A. Crere, nd A. Ambinis,, Qunum wlks driven by mny coins Phys. Rev. A 67, qun-ph/6 4. J. Kempe, 3 Qunum wlks- n inroducory overview, Conemporry Physics , qun-ph/ F. Sruch, 7 Connecing he discree nd coninuous ime qunum wlks,qun-ph/ J. R. Busemeyer nd J. T. Townsend 4 Qunum dynmics of humn decision mking Jnl. of Mh. Psych. 7. M. Lee, I. Fuss nd D. Nvrro 6 A Byesin Approch o Diffusion Models of Decision-Mking nd Response Time NIPS 8. L. K. Grover 997 Phys. Rev. Le S. Nguleswrn, I. Fuss, nd L. Whie 6, Auomed plnning using qunum compuion, Proc. of ICAPS 6. Y. Ahrnov, L. Dvidovich, nd N. Zgury, 99 Qunum Rndom wlks, Phys. Rev. A D. A. Meyer, 996, From qunum cellulr uom o lice gses, J. S. Phys , qun-ph/9643. See procedures from solid se physics e. g. J. Zimn 97 Principles of he Theory of Solids Cmbridge 3. M. A. Neilsen nd I. L. Chung Qunum Compuion nd Qunum Informion Cmbridge 4. E. Merzbcher 998 Qunum Mechnics Wiley 3 rd Ediion 5. G. B. Arfken nd H. J. Weber 5 Mhemicl Mehods for Physiciss Elsevier 5 h Ediion 6. G. Cssell nd R Berger 99 Sisicl Inference Wdsworh nd Brooks/Cole Appendi A The ype II Chebyshev polynomils obey he recursion Hence U U ( ) U ( ) U ( ) =. (A) n+ n n ip ip ( ( p) ) ( e + e ) U cos( p) ( ) U ( cos( p) ) + cos =. (A) The inverse Fourier rnsform of his relion is he recursion Uˆ ( : ) Uˆ ( : + ) + Uˆ ( : ) Uˆ ( ) =. (A3) + :

23 Thus from equion 55 in he min e we obin he recursion for he polynomils P ( ) P ( ) + P ( ) P ( ) =. (A4) + k k + k k