We derive the momentum space dynamic equations and state functions for one dimensional quantum walks by using linear systems and Lie group theory. The momentum space provides an analytic capability similar to that contributed by the z transform in discrete systems theory. The state functions at each time step are expressed as a simple sum of three Chebyshev polynomials. The functions provide an analytic expression for the development of the walks with time.Ian Fuss, Langord B. White, Peter J. Sherman, Sanjeev Naguleswara

Fuss, Ian

Naguleswaran, Sanjeevan

Sherman, Peter J.

White, Langford Barton

arXiv

Adelaide Research &amp; Scholarship

arXiv:quant-ph/0604197v2  24 May 2006Momentum Dynamis of One Dimensional Quantum WalksIan Fuss∗Defene Siene and Tehnology Organisation (DSTO), Edinburgh, Australia. andShool of Eletrial and Eletroni Engineering, University of Adelaide, Australia.Langford B. White and Sanjeev NaguleswaranShool of Eletrial and Eletroni Engineering, University of Adelaide, Australia.Peter J. ShermanDepartment of Aerospae Engineering, Iowa State University, Ames, Iowa.We derive the momentum spae dynami equations and state funtions for one dimensional quan-tum walks by using linear systems and Lie group theory. The momentum spae provides an analytiapability similar to that ontributed by the z transform in disrete systems theory. The statefuntions at eah time step are expressed as a simple sum of three Chebyshev polynomials. Thefuntions provide an analyti expression for the development of the walks with time.I. INTRODUCTIONThe study of quantum walks has reeived onsiderable attention sine the introdutory papers on the subjet, suhas [1, 2℄ and referenes therein. In this paper, we develop an analyti approah to study the properties of these walksbased on a momentum spae representation.This paper is strutured suh that in Setion 2 of the paper the momentum spae dynami equations for one dimen-sional quantum walks are derived via the Z transform of the position spae dynami equations and its representationof the disrete Fourier transform when Z lies on the unit irle. An exponential form of of the momentum spae timeoperator is derived in setion 3 by using the group theory of SU(2) and a matrix inner produt spae. The exponentialform allows a simple analyti alulation of the time evolution operator for arbitrary time intervals. This is used inSetion 4 to obtain analyti expressions for the momentum spae wave funtions of quantum walks at arbitrary times.These wave funtions are expressed quite simply in terms of Chebyshev Polynomials of the seond kind. Some plotsof the momentum spae probability densities for dierent parameter values and times are provided in setion 5. Theonlusions are summarised in Setion 6.II. MOMENTUM SPACE DYNAMIC EQUATIONSFor a given ψ(0, 0) we onsider the evolution of a quantum state ψ(t, x) ∈ C2 for disrete times t ≥ 0 on a linex ∈ Z. The dynamis of the state then evolve aording to the dierene equations,ψ0(t, x) = eiα[aψ0(t− 1, x− 1) + bψ1(t− 1, x− 1)],ψ1(t, x) = eiα[−b∗ψ0(t− 1, x+ 1) + a∗ψ1(t− 1, x+ 1)], (1)where |a|2 + |b|2 = 1 and α ∈ R.Taking two-dimensional Z transforms of these equations yieldsψ0(z1, z2) = eiαz−11 z−12 [aψ0(z1, z2) + bψ1(z1, z2)ψ1(z1, z2) = eiαz−11 z−12 [−b∗ψ0(z1, z2) + a∗ψ1(z1, z2). (2)Thus the transfer matrix for the system isB(z1, z2) = eiαz−11[az−12 bz−12−b∗z2 a∗z2](3)∗Eletroni address: Ian.Fussdsto.defene.gov.au2therefore, for any iteration (time) index n, the quantum walk state Ψ(n, x) has transform x↔ zΨ(n, x)↔ einαCn(z)Ψ(0, 0), (4)where C(z) is the matrix polynomialC(z) =[az−1 bz−1−b ∗ z a ∗ z]. (5)It should be noted that C is paraunitary, that is C−1(z) = CT (1/z). In partiular this implies that C(z) is unitaryon |z| = 1. Further we note that detC(eip)) = 1 and hene the matrixS(p) = C(eip) (6)is unimodular. The Fourier transform x↔ p isΨ(n, x)↔ einαSn(p)Ψ(0, 0). (7)Thus by hoosing Plank's onstant ~ = 1, the momentum spae representation of the quantum walk state vetorφ(n, p) evolves asφ(n, p) = einαSn(p)φ(0, p), (8)whereφ(0, p) = ψ(0, 0) =[ψ0(0, 0)ψ1(0, 0)]. (9)Thus the time evolution operator in the momentum spae is a 2 × 2 matrix polynomial. Hene, the momentumspae equations are muh more amenable to analysis than those in position spae.III. EXPONENTIATION OF THE TIME EVOLUTION OPERATORThe unimodular matrix S(p) an be written in exponential form asS(p) = Exp(iθ(p)−→c (p).−→σ ) (10)where θ and −→c are real funtions of p and the matrix vetor −→σ has Pauli matrix omponents [3℄σ1 =[0 11 0],σ2 =[0 −ii 0]andσ3 =[1 00 −1]. (11)The inner produt(A,B) =12Tr(AB)dened on the vetor spae of 2×2 unitary matries gives an inner produt spae. The set of matries {I, σ1, σ2,σ3},provide an ortho-normal basis for this spae.The oeients of the matries an be evaluated by taking the inner produt of both sides of (10)(σi, S(p)) = (σi, Exp(iθ(p)−→c (p).−→σ )3with eah of the matries σi. In doing this we note that a generalised de-Moivre priniple givesExp(iθ−→c .−→σ ) = Icos(θ) + i−→c .−→σ sin(θ),where the p dpendene has been suppressed for simpliity. Hene,(I, Exp(iθ−→c .−→σ )) = cos(θ) (12)and(σj , Exp(iθ−→c .−→σ )) = icjsin(θ). (13)The equivalent oeients for S(p) an be obtained by deninga = cos(β)e−iγ ,b = sin(β)e−iδ. (14)Substituting in (6) givesS(p) =[cos(β)e−i(p+γ) sin(β)e−i(p+δ)−sin(β)ei(p+δ) cos(β)ei(p+γ)]. (15)These expressions an be simplied by setting p′ = p + γ and p′′ = p + δ. Using de Moivre's priniple one againwe obtain the transition matrix oeients(I, S(p)) = cos(β)cos(p′),(σ1, S(p)) = −isin(β)sin(p′′),(σ2, S(p)) = isin(β)cos(p′′),(σ3, S(p)) = −icos(β)sin(p′). (16)Comparing oeients in equations (12) and (13) with those of (16) we obtaincos(θ) = cos(β)cos(p′),c1sin(θ) = −sin(β)sin(p′′),c2sin(θ) = sin(β)cos(p′′),c3sin(θ) = −cos(β)sin(p′). (17)IV. MOMENTUM SPACE STATE FUNCTIONSA dynami equation for momentum spae state funtions was given in (8). The exponentiation of the operator in(10) enables us to write the powers of the evolution operator asSn(p) = Exp(inθ−→c .−→σ ) = Icos(nθ) + i−→c .−→σ sin(nθ). (18)The trigonometri expressions in the above equation an be expressed in terms of the Chebyshev polynomials Tnand Un as [4℄cos(nθ) = Tn(cos(θ))andsin(nθ) = Un−1(cos(θ))sin(θ). (19)4Using these expressions and writing the dot produt as a sum of omponents (11) beomesSn(p) = Tn(cos(θ))I + iUn−1(cos(θ))3∑i=1cisin(θ)σi. (20)The equalities of (17) enable us to rewrite this asSn(p) = Tn(cos(β)cos(p′))I − iUn−1(cos(β)cos(p′))[sin(β)sin(p′′)σ1 − sin(β)cos(p′′)σ2 + cos(β)sin(p′)σ3] (21)Using the Pauli matries the matrix polynomialSn(p) =[Tn(cos(β)cos(p′) Un−1(cos(β)cos(p′))sin(β)cos(p′′)−Un−1(cos(β)cos(p′))sin(β)cos(p′′) Tn(cos(β)cos(p′))]− i[Un−1(cos(β)cos(p′))cos(β)sin(p′) Un−1(cos(β)cos(p′))sin(β)sin(p′′)Un−1(cos(β)cos(p′))sin(β)sin(p′′) −Un−1(cos(β)cos(p′))cos(β)sin(p′)](22)is obtained.The evolution of the quantum walk in momentum spae representation given in (8 )an also be expressed asφ(n, p)e−inα = Sn(p)φ(0, p). (23)(22) and (9) enable this expression to be written asφ0(n, p)e−inα = [Tn(cos(β)cos(p′))− iUn−1(cos(β)cos(p′))cos(β)sin(p′)]Ψ0(0, 0)+[Un−1(cos(β)cos(p′))sin(β)cos(p′′)− iUn−1(cos(β)cos(p′))sin(β)sin(p′′)]Ψ1(0, 0) (24)φ1(n, p)e−inα = −[Un−1(cos(β)cos(p′))sin(β)cos(p′′) + iUn−1(cos(β)cos(p′))sin(β)sin(p′′)]Ψ0(0, 0)+Tn(cos(β)cos(p′)) + iUn−1(cos(β)cos(p′))cos(β)sin(p′)]Ψ1(0, 0) (25)By using the relationTn(x) = Un(x) − xUn−1(x) (26)this an be written asφ0(n, p)e−inα = [Un(cos(β)cos(p′))− Un−1(cos(β)cos(p′))cos(β)[cos(p′) + isin(p′)]]Ψ0(0, 0)+[[Un−1(cos(β)cos(p′))sin(β)[cos(p′′)− sin(p′′)]]Ψ1(0, 0) (27)φ1(n, p)e−inα = −[Un−1(cos(β)cos(p′))sin(β)[cos(p′′) + isin(p′′)]]Ψ0(0, 0)+[Un(cos(β)cos(p′))− Un−1(cos(β)cos(p′))cos(β)[cos(p′) + isin(p′)]]Ψ1(0, 0). (28)Inverting the de Moivre formula and moving the global phase term to the right hand side gives the analytiexpressionsφ0(n, p) = einα[Un(cos(β)cos(p′))− Un−1(cos(β)cos(p′))cos(β)eip]Ψ0(0, 0)+einα[Un−1(cos(β)cos(p′))sin(β)e−ip]Ψ1(0, 0) (29)φ1(n, p) = −einα[Un−1(cos(β)cos(p′))eip]Ψ0(0, 0)+einα[Un(cos(β)cos(p′))− Un−1(cos(β)cos(p′))cos(β)e−ip]Ψ1(0, 0) (30)for the general momentum spae state funtions for a one dimensional quantum walk at time n.5−4 −2 0 2 40.20.40.60.81densityT=10−4 −2 0 2 40.20.40.60.81densityT=30−4 −2 0 2 400.20.40.60.81pdensityT=50−4 −2 0 2 400.20.40.60.81pdensityT=70Figure 1: Momentum Spae Density funtions for β = pi8−4 −2 0 2 400.20.40.60.81densityT=10−4 −2 0 2 400.20.40.60.81densityT=30−4 −2 0 2 400.20.40.60.81pdensityT=50−4 −2 0 2 400.20.40.60.81pdensityT=70Figure 2: Momentum Spae Density funtions for β = pi4V. MOMENTUM SPACE DENSITIESThe denisity |φ0(p : t)|2for α = γ = δ = 0, Ψ0(0, 0) = 1 and Ψ1(0, 0) = 0 is plotted in gures 1, 2, 3 for β =pi8 ,pi4and3pi8 and for times t = 10, 30, 50, 70. When β is xed the dominant feature of the time series is an inrease inosillation frequeny with time. This orresponds to the inrease in support of the position spae densities with time.The eet of inreasing β is to trade a derease in the onstant omponent of the density funtion for an inreasein the osillatory omponent. This orresponds to a shift in the position spae of probability density from the zeroregion of the walk to the outer edges of the walk.The sequenes shows that the densities onverge to a limit as time inreases. They also illustrate the fat that6−4 −2 0 2 400.20.40.60.81densityT=10−4 −2 0 2 400.20.40.60.81densityT=30−4 −2 0 2 400.20.40.60.81pdensityT=50−4 −2 0 2 400.20.40.60.81pdensityT=70Figure 3: Momentum Spae Density funtions for β = 3pi8the momentum spae is an attrative representation in whih to derive this limit beause the domain of the wavefuntions is onstant, p ∈ [−pi, pi]. This is in ontrast to the real spae where the domain expands with time.VI. CONCLUSIONSIt has been shown that the momentum spae dynami equations for a quantum walk an be derived using a ztransform of the position spae equations for the dynami walk. An exponential representation of the momentum spaetime evolution operator was derived by using Lie group theory. This enabled the alulation of general momentumspae wave funtions in terms of Chebyshev polynomials. Some simple alulations of the momentum spae probabilitydensities illustrate the onvergene of the momentum wave funtions to a limit as time inreases.[1℄ D. Aharanov, A. Ambainis, J. Kempe, U. Vazirani. Quantum walks on graphs. arXiv:quant-ph/0012090, 2000.[2℄ J. Kempe. Quantum walks - an introdutory overview. Contemporary Physis and arXiv:quant-ph/0303081v1, 44:307327,2003.[3℄ E. Merzbaher. Quantum Mehanis. Wiley, 1998.[4℄ G. Arfken, H. Weber. Mathematial Methods for Physiists. Elsevier, 2005.

Momentum dynamics of one dimensional quantum walks

https://digital.library.adelaide.edu.au/dspace/bitstream/2440/43692/1/hdl_43692.pdf

Momentum dynamics of one dimensional quantum walks

Abstract

Similar works

Full text

Available Versions

Adelaide Research & Scholarship

Momentum dynamics of one dimensional quantum walks

Abstract

Similar works

Full text

Available Versions

Adelaide Research &amp; Scholarship

Adelaide Research & Scholarship