The Heun differential equation and the Gauss differential equation related to quantum walks
aa r X i v : . [ qu a n t - ph ] A p r The Heun differential equation and the Gaussdifferential equation related to quantum walks
Norio Konno · Takuya Machida · TohruWakasaAbstract
The limit theorems of discrete- and continuous-time quantum walkson the line have been intensively studied. We show a relation among limitdistributions of quantum walks, Heun differential equations and Gauss dif-ferential equations. Indeed, we derive the second-order Fucksian differentialequations which limit density functions of quantum walks satisfy. Moreover,using both differential equations, we discuss a relationship between discrete-and continuous-time quantum walks. Taking suitable limit, we can transforma Heun equation obtained from the limit density function of the discrete-timequantum walk to a Gauss equation given by that of the continuous-time quan-tum walk.
The discrete-time quantum walk (QW), which is a quantum counterpart ofthe classical random walk, has been extensively investigated since Ambainiset al. [2] studied a detail of the walk. The continuous-time QW was proposedby Farhi and Gutmann [5], and has been analyzed on not only regular graphsbut also complex networks (e.g. [8,19,32]). A further development in the theoryof the QW during recent 10 years showed us several novel properties on bothQWs. The behavior of the QW is quite different from that of the randomwalk and is expected to be connected with various phenomena in quantum · Heun differential equation · Gauss differentialequation.
Acknowledgements
N. K. was partially supported by the Grant-in-Aid for Scientific Re-search (C) of Japan Society for the Promotion of Science (Grant No. 21540118). T. M. and T.W. are grateful to the Meiji University Global COE Program “Formation and Developmentof Mathematical Sciences Based on Modeling and Analysis” for the support. mechanics. A relation between the QW and quantum computer has been alsodiscussed (e.g. [1,20,23]). The quantum search algorithm designed by the QWis one of the important applications. The Grover search algorithm can beconsidered as a discrete-time QW on the complete graph and it producesspeed-up at the square-root rate for the corresponding classical search. Thereare some reviews of the QW [10,11,15,31].One of the purposes in the study of the QW is to derive the limit dis-tribution and the asymptotic behavior as time tends to infinity. The limitdistribution has been obtained for various kinds of QWs. In the present paper,we concentrate on the QW in the one-dimensional space. Observing previousresults, one can mention that each limit distribution usually has a compactsupport and admits a singularity at the boundary of the support. This is aninteresting property of the limit distribution, because the QW does not pos-sess any singularity in space. To the fact, we now address a question: How tounderstand the singularity of the QW? Motivated by this question, we inves-tigate the limit distribution of the QW from a viewpoint of the differentialequation.We begin with finding a differential equation for the limit distribution.For example, consider discrete- and continuous-time symmetric simple randomwalks on the line. The central limit theorem for the walk, that is, lim t →∞ P ( Y t / √ t ≤ x ) = R x −∞ e − y / / √ π dy , is well-known, where Y t denotes the walker’s posi-tion at time t and P ( Y t = x ) is the probability that the walker is at position x at time t . For the limit density function f ( x ) = e − x / / √ π , an equation d f ( x ) /dx + xdf ( x ) /dx + f ( x ) = 0 is derived. This equation comes from thetheory of the diffusion equation ∂u ( x, t ) /∂t = ∂ u ( x, t ) /∂x , and in par-ticular, it is also obtained by the self-similarity of the fundamental solution u ( x, t ) = (1 / √ t ) f ( x/ √ t ).Let us consider both discrete-time and continuous-time QWs on the line.The classical random walk corresponds to the diffusion process, while for theQW the corresponding process is not known. Thus, to find out the processwould remain as one of important problems. Taking an account of the singu-larity in the limit distribution, we treat a class of Fucksian linear differentialequations of the second order. For discrete-time (resp. continuous-time) QWs,we are led to a Heun’s differential equation (HE) (resp. a hypergeometric dif-ferential equation by Gauss (the Gauss equation, GE)). The GE is one oftypical Fucksian equations and represents the Fucksian equations with exactlythree regular singular points. On the other hand, the HE was proposed in 1888by Heun [7] and admits four regular singular points.Moreover, a concept of confluence between the HE and the GE helps us tounderstand a relationship between discrete- and continuous-time QWs clearly.Through these results, authors believe that the HE and the GE play an im-portant role in understanding the QWs.To this end, we will mention a significant remark on the QW and the BC Inozemtsev model, which is an integrable quantum system. These twomodels are to be connected through HEs. Therefore, we would like to expect a new mathematical theory on the relation between QWs, HEs, and quantummechanics.The present paper is organized as follows. In Sect. 2, we explain both theone-dimensional discrete-time QW and the HE. After introducing the limitdistribution obtained by Konno [12,14], we discuss a relation between thediscrete-time QW and the HE. In Sect. 3, we concentrate on the continuous-time QW, and show that the GE relates with the limit density function ofthe walk. Section 4 is devoted to a connection between the HE and the GEin order to understand a relationship between discrete- and continuous-timeQWs given by Strauch [24]. In the final section, we summarize the relationsobtained here and propose a future problem on our results. Furthermore, wetransform the HE to the BC Inozemtsev system in Appendix 1 and showrelations among measure of the QW, the HE and the GE in Appendix 2.
In this section, we discuss a relation between the discrete-time QW in onedimension and the HE. At first we define the QW on the line. Let | x i ( x ∈ Z = { , ± , ± , . . . } ) be infinite components vectors which denote the position ofthe walker. Here, x -th component of | x i is 1 and the other is 0. Let | ψ t ( x ) i ∈ C be the amplitude of the walker at position x at time t ∈ { , , , . . . } , where C is the set of complex numbers. The amplitude of the walk at time t is expressedby | Ψ t i = X x ∈ Z | x i ⊗ | ψ t ( x ) i . (1)The time evolution of the walk can be defined by the following unitary matrix: U = (cid:20) a bc d (cid:21) , (2)where a, b, c, d ∈ C . Moreover, we introduce two matrices: P = (cid:20) a b (cid:21) , Q = (cid:20) c d (cid:21) . (3)Note that P + Q = U . Then the evolution is determined by | ψ t +1 ( x ) i = P | ψ t ( x + 1) i + Q | ψ t ( x − i . (4) The probability that the discrete-time quantum walker X ( d ) t is at position x at time t , P ( X ( d ) t = x ), is defined by P ( X ( d ) t = x ) = h ψ t ( x ) | ψ t ( x ) i , (5)where h ψ t ( x ) | denotes the conjugate transposed vector of | ψ t ( x ) i . In the presentpaper, we take the initial state as | ψ ( x ) i = (cid:26) T [ 1 / √ , i/ √ x = 0) , T [ 0 , x = 0) , (6)where T is the transposed operator. Equation (6) is well-known as the initialstate that gives the symmetric probability distribution about the origin [15].For the walk, some limit theorems and asymptotic behaviors have beenobtained. In particular, we focus on the density function of probability distri-bution as t → ∞ . The limit distribution of the QW was given by Konno [12,14]. That is, for abcd = 0, we havelim t →∞ P X ( d ) t t ≤ x ! = Z x −∞ f ( d ) ( y ) I ( −| a | , | a | ) ( y ) dy, (7)where f ( d ) ( x ) = p − | a | π (1 − x ) p | a | − x , (8)and I A ( x ) = 1 if x ∈ A , I A ( x ) = 0 if x / ∈ A . A lot of limit distribu-tions of discrete-time QWs are often described by using this function f ( d ) ( x )(e.g. [3,9,16,17,18,22]). Figure 1 depicts the comparison between the proba-bility distribution at time 500 and the limit density function for the walk with a = b = c = − d = 1 / √
2, which is called the Hadamard walk. x (a) d e n s i t y f un c t i o n -1 1 0 x (b) Fig. 1
Comparison between the probability distribution and the limit density function ofthe discrete-time QW with a = b = c = − d = 1 / √
2. Figure (a) is the probability distributionat time t = 500. Figure (b) is the limit density function of the walk. From now, we discuss a relation between the limit density function f ( d ) ( x )and the HE. The canonical form of the HE is given by d udz + (cid:18) γz + δz − ǫz − θ (cid:19) dudz + αβz − qz ( z − z − θ ) u = 0 , (9)where θ ∈ C is one of the singular points and α, β, γ, δ, ǫ, q ∈ C . The fiveparameters α, β, γ, δ, ǫ are linked by the relation α + β + 1 = γ + δ + ǫ andthe parameter q is called the accessory parameter. The HE has four regularsingularities at z = 0 , , θ, ∞ . Particularly the HE with γ = δ = ǫ = 1 / θ = 1 , q = αβ case).We find that f ( d ) ( x ) satisfies the following differential equation:(1 − x )( | a | − x ) d dx f ( d ) ( x ) − x (4 | a | + 3 − x ) ddx f ( d ) ( x )+(9 x − | a | − f ( d ) ( x ) = 0 . (10)By a change of an independent variable x = z , we obtain one of our mainresults: Theorem 1 d dz u ( d ) ( z ) + (cid:18) z + 2 z − z − | a | (cid:19) ddz u ( d ) ( z )+ z − | a | +14 z ( z − z − | a | ) u ( d ) ( z ) = 0 , (11) where u ( d ) ( z ) = p − | a | π (1 − z ) p | a | − z . (12)We should remark that Eq. (11) is equivalent to the HE with α = β = 32 , γ = 12 , δ = 2 , ǫ = 32 , q = 2 | a | + 14 , θ = | a | . (13) In this section we will show a relationship between the continuous-time QWon Z and the GE: z ( z − d udz + { ( α + β + 1) z − γ } dudz + αβu = 0 (14)where α, β, γ ∈ C are parameters. Equation (14) is a second-order Fucksianequation with three regular singularities at z = 0 , , ∞ .We give a definition of the continuous-time QW on the line. At first, weconsider the amplitude ψ t ( x ) ∈ C at position x ∈ Z at time t ( >
0) instead of | ψ t ( x ) i ∈ C for the discrete-time QW. The evolution of the amplitude isdefined by i dψ t ( x ) dt = − ν { ψ t ( x − − ψ t ( x ) + ψ t ( x + 1) } , (15)where ν >
0. The probability that the walker is at position x at time t isdenoted by P ( X ( c ) t = x ) = | ψ t ( x ) | , where X ( c ) t is the continuous-time quan-tum walker’s position at time t . We take an initial state as ψ (0) = 1 and ψ ( x ) = 0 ( x = 0). Konno [13] and Gottlieb [6] got the limit theorem for thecontinuous-time QW as follows:lim t →∞ P X ( c ) t t ≤ x ! = Z x −∞ f ( c ) ( y ) I ( − ν, ν ) ( y ) dy, (16)where f ( c ) ( x ) = 1 π p (2 ν ) − x . (17)In Fig. 2, we show the comparison between the probability distribution at time500 and the limit density function for the walk with ν = 1 / √ x (a) -1 0 1 d e n s i t y f un c t i o n x (b) Fig. 2
Comparison between the probability distribution and the limit density function ofthe continuous-time QW with ν = 1 / √
2. Figure (a) is the probability distribution at time t = 500. Figure (b) is the limit density function of the walk. We see that the function f ( c ) ( x ) satisfies the following differential equation:(4 ν − x ) d dx f ( c ) ( x ) − x ddx f ( c ) ( x ) − f ( c ) ( x ) = 0 . (18)By the change of an independent variable x / ν = z , we have a continuous-time counterpart of Theorem 1: Theorem 2 z ( z − d dz u ( c ) ( z ) + (cid:18) z − (cid:19) ddz u ( c ) ( z ) + 14 u ( c ) ( z ) = 0 , (19) where u ( c ) ( z ) = 12 νπ √ − z . (20)Note that Eq. (19) is the GE with α = β = γ = 1 / In this section, we discuss a relation between discrete- and continuous-timeQWs on the second-order differential equations which were obtained in previ-ous sections. Strauch [24] found a relation between both walks by transformingEq. (4) to Eq. (15) in a suitable limit. D’Alessandro [4] focused on both QWson general graphs and derived the dynamics of continuous-time walks as alimit of discrete-time dynamics.Considering a confluent type for the HE which the function u ( d ) ( z ) satisfies,we can also obtain the GE which the function u ( c ) ( z ) satisfies. For a scalingparameter τ , substituting z = t/τ in Eq. (11), we have4 t ( | a | τ − t ) (cid:18) − tτ (cid:19) d dt v ( t ) + 2 (cid:26) t τ − (5 | a | + 4) t + | a | τ (cid:27) ddt v ( t )+ (cid:18) tτ − | a | − (cid:19) v ( t ) = 0 , (21)with v ( t ) = u ( d ) ( t/τ ). As τ → ∞ , | a | → | a | τ →
1, weobtain a confluent HE: t ( t − d vdt + (cid:18) t − (cid:19) dvdt + 14 v = 0 . (22)Equation (22) is equivalent to the GE which was obtained from the continuous-time QW (see Eq. (19)). This result corresponds to the result in Strauch [24].We can confirm his result in a relation between the HE and the GE. In this section, we discuss and conclude our results. In the present paper,we found that for the discrete-time QW, the limit density function f ( d ) ( x )satisfies a HE, while for the continuous-time QW, the limit density function f ( c ) ( x ) satisfies a GE. Moreover, considering the confluent HE, we confirmeda relation between the discrete- and the continuous-time QWs correspondingto the result by Strauch [24].As a significant remark, we would like to mention a relationship between thediscrete-time QW and a Schr¨odinger equation through the HE (see Appendix 1for detail). The Hamiltonian of the continuous-time QW is given by the adja-cency matrix of the graph on which the walk is defined [5]. However the Hamil-tonian of the discrete-time QW is not known. On the other hand, it was shown by Takemura [25,26,27,28,29,30] that the HE can be transformed to the BC Inozemtsev system which is a one-particle integrable quantum system. In ad-dition, the BC Inozemtsev system includes the Calogero-Moser-Sutherlandsystem or the Olshanetsky-Perelonov system [21]. And the Hamiltonian is ex-pressed with the Weierstrass ℘ -function. Thus, to find a relation among thediscrete-time QW, the HE and the BC Inozemtsev system might be one ofthe interesting problems.
Appendix 1 A relation between the discrete-time QW and the BC Inozemtsev model
In this Appendix, we transform the HE obtained in Sect. 2 to the BC Inozemtsev system.The BC Inozemtsev model is known as an integrable one-particle quantum system includingthe Calogero-Moser-Sutherland system (see [25,26,27,28,29,30]). The Hamiltonian H of the BC Inozemtsev model is defined by H = − d dx + l j X i =0 ( l j + 1) ℘ ( x + w j ) , (23)where the function ℘ ( x ) is the Weierstrass ℘ -function with periods (2 w , w ) and theparameters l j ∈ C ( j = 0 , , ,
3) are constants. We should note that w , w ∈ C are linearlyindependent on R , where R is the set of real numbers. By using the following transformationfor Eq. (11) (see Takemura [27,28] for detail), z = ℘ ( x ) − ℘ ( w ) ℘ ( w ) − ℘ ( w ) (24)and putting g ( x ) = u ( d ) ( z ) z / ( z − | a | ) / , we can get Hg ( x ) = 2 − | a | { ℘ ( w ) − ℘ ( − w − w ) } g ( x ) , (25)where H = − d dx − ℘ ( x ) + 34 ℘ ( x − w − w ) + 2 ℘ ( x + w ) . (26)The operator H is the Hamiltonian of the BC Inozemtsev model with l = − , l = 0 , l = − , l = 1 , w = 0 , w = − w − w , where w , w are arbitrary non-zero complex numbersand linearly independent on R . Therefore the eigenvalue of the Hamiltonian H determinedby Eq. (26) is −| a | { ℘ ( w ) − ℘ ( − w − w ) } and the eigenfunction corresponding to theeigenvalue is the above mentioned function g ( x ). Via the HE, we found that the limit densityfunction f ( d ) ( x ) of the discrete-time QWs is related to the BC Inozemtsev model.
Appendix 2 Relations among measure of the QW, the HE and theGE
In this Appendix, we focus on both limit measures f ( d ) ( x ) dx and f ( c ) ( x ) dx and discuss arelation among the QW, the HE and the GE. Putting x = z for the measure f ( d ) ( x ) dx ofthe discrete-time QW, we get f ( d ) ( x ) dx = √ −| a | π (1 − z ) √ | a | z − z dz ( x > , − √ −| a | π (1 − z ) √ | a | z − z dz ( x < . (27)The function w ( d ) ( z ) = p − | a | / π (1 − z ) p | a | z − z satisfies the HE with α = β =2 , γ = , δ = 2 , ǫ = , q = | a | +22 , θ = | a | : d dz w ( d ) ( z ) + z + 2 z − z − | a | ! ddz w ( d ) ( z ) + 4 z − | a | +22 z ( z − z − | a | ) w ( d ) ( z ) = 0 . (28)In the case of the continuous-time QW, putting x / ν = z for f ( c ) ( x ) dx , we obtain f ( c ) ( x ) dx = νπ √ z − z dz ( x > , − νπ √ z − z dz ( x < . (29)For the function w ( c ) ( z ) = ν/π √ z − z , the following GE with α = β = 1 , γ = is realized: z ( z − d dz w ( c ) ( z ) + (cid:18) z − (cid:19) ddz w ( c ) ( z ) + w ( c ) ( z ) = 0 . (30)Note that we can derive Eq. (30) from Eq. (28) in a similar fashion as in Sect. 4. More-over Eq. (28) can be transformed into the eigen equation of the BC Inozemtsev modelwith l = − , l = − , l = − , l = − , w = 0 , w = − w − w and the eigenvalue −| a | { ℘ ( w ) − ℘ ( − w − w ) } . References
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