Solution of the Time Dependent Schrödinger Equation and the Advection Equation via Quantum Walk with Variable Parameters

We propose a solution method of Time Dependent Schrödinger Equation (TDSE) and the advection equation by quantum walk/quantum cellular automaton with spatially or temporally variable parameters. Using numerical method, we establish the quantitative relation between the quantum walk with the space dependent parameters and the “Time Dependent Schrödinger Equation with a space dependent imaginary diffusion coefficient” or “the advection equation with space dependent velocity fields”. Using the 4-point-averaging manipulation in the solution of advection equation by quantum walk, we find that only one component can be extracted out of two components of left-moving and right-moving solutions. In general it is not so easy to solve an advection equation without numerical diffusion, but this method provides perfectly diffusion free solution by virtue of its unitarity. Moreover our findings provide a clue to find more general space dependent formalisms such as solution method of TDSE with space dependent resolution by quantum walk.


Introduction
Quantum walk is a mechanical system evolved by a discrete local unitary transformation and is regarded as a quantum version of the classical random walk [1,2].In recent years, relations between the quantum walk and continuous quantum wave equations were shown [3][4][5][6][7][8][9] and more recently a certain view point was added to the relation between the quantum walk and the Time Dependent Schrödinger Equation (TDSE) [10].
There are some formalisms on the broadly-defined quantum walk.Here we use the formalism usually called as quantum cellular automaton (QCA).Namely, we simply consider it as a mechanical system evolved by a unitary transformation by a banded matrix.
A quantum walk can be most easily conceived by comparing it with classical random walk.One-dimensional classical random walk is defined by a transition probability matrix applying a probability distribution ˆij The conservation of probability requires And the requirement that between neighboring grid points means that should be e l the transition occurs only ˆij P a banded matrix.In quantum walk, where probability amplitud evolves instead, this transition probability matrix is rep aced by the unitary banded matrix.Namely, Incidentally the difference between a random and a quantum process is that in the latter case the proba process bility is given by squaring the amplitude.In order to conserve total probability, the transformation must be unitary, While it is apparently easy to make some transition f probability ( (1.2)), it requires some devices to con probability matrix that satisfies the conservation o Equation struct a unitary banded matrix.However we note that the unitary banded matrix represented on discrete space is quite similar to the twoscale transformation matrix in an orthogonal wavelet with compact support [11].
One of the easiest way to construct the unitary banded matrix is to use the product of trivial unitary banded matrices as shown in Figure 1.Here we refer to a block diagonal matrix whose block diagonal components are 2 × 2 unitary matrices as a trivial unitary banded matrix.
By using the product of trivial unitary matrices in the RHS, we can construct a non-trivial unitary banded matrix in the LHS of Figure 1.In fact, reversely it is true that any translationally invariant unitary banded matrix can be factorized in the form of the RHS (see Appendix B).
In general we can introduce space dependent parameters in the quantum walk (see Here we use U in place of E and F in Figure 1. In these parameters, b(x) means potential te TDSE (see Appendix A).
is a local gauge transformation pa amely redefinition of the phase of wave functio rameter (n n), but we don't discuss space dependent   x  here while it is an interesting subject.

Time Dependent Schrödinger Equation with Space Dependent Imaginary Diffusion Coefficient
Here we discuss space dependent parameters pects to space x are first and second derivatives with res and .D  We show some examples as follows.
H as a basis for which theoretical solution can be obtained easily, H  can be rewritten as ona ges We must note that additi l potential term emer when  is changed.
H  For the case of H as the linear combination of , we have more general form Since analytical derivationof ,    is not straightforwar because of the d broken tra slational invariance, w n e determine ,    using a numerical method.Namely, the solution for and compared with the analytical solution for We determine ,    so that the two solutions completely coincide.
The used 2 × 2 matrix of quantum , the first term in   b x is crucial and leads to m ss results without it).In our numerical method, we use periodic boundary condition for the range [0,1], and use two types of A(x).
(More precisely, in the actual calculation the range [0,1] was mapped to the range [0,512]) 1) sine function type eaningle where 4K(k) is the period along the real axis of elliptic function s 0 2π 4 4 , In order to calculate the theoretical solution for we can simply reduce it to the solution of the free fields TD (Note that as , the periodicity     , t x is guaranteed).nction forms for the B( for sine function type and elliptic function type respectiv We use the fact olution of the free fie n be written [12] as ely.
that theoretical s ld TDSE for the Gaussian wave packet ca 

 
To summarize these results, we find that by selecting only two parameters 1 4, 1 8 , tum walk num this result leads to 0   .
Namely we find that is the right Hamiltonian for the continuum limit evolution equation corresponding to the TDSE-type quantum walk with space dependent   x  .

Advection Equation with Space Dependent Velocity Field
We consider space dependentparameters     in the advection-type quantum walk (see Appendi Here we use

 
U x in place of E and F in Figure 1 and The evolution equation that emerges as the continuum limit for the advection-type quantum walk with space dependent velocity field 2) a theoretical solution and a quan erical solution coincides completely.
where However since quantum walk is a unitary transformation, only the unitary-type advection equation is allowed.
In the following, w ve e in stigate using a numerical ion indeed e periodic boundary follow (More precisely, in the actual calculation the range [0,1] was mapped to the range [0,512]) method if the unitary-type advection equat emerges as a solution for continuum limit.
In our numerical method, we us condition for the range [0,1], and use the ing type of A(x).
In order to calculate the theoretical solution for we can simply reduce it to the solution of the constant velocity field (   1 We used for the initial wave packet.In fact, in the case of advection type quantu both left moving and right moving components emerge. that using the 4-point-averaging maniract only the one component (see Appendix C).
4-point-averaging is an aver four neighboring grid points in space-time.m walk, However we find pulation, we can ext aging manipulation over    and therefore  


We show the numerical solutions in Figures 6-8 using the above prescriptions.
To summarize, from the numerical examination we find that method 2 is the right prescription.0, : 0, , he mathematics of quantum walk with variable parameters is not well established and difficult to derive space-time equation for its continuum limit by purely In general it is not so easy to solve an advection equation without numerical diffusion, but this method provides perfectly diffusion free solution by virtue of its unitarity.In the present work, we employ QCA formalism where extended space generated by combining original physical space and coin space (internal degree of freedom) is used.

Conclusion
On the extended discrete space, mathematics of quantum walks becomes more clear.
The extension to the multidimensional space is straightforward and currently we are applying the methodology to more realistic inhomogeneous quantum system in orhis research was supported in part by TUT Programs on eories with numerical methods especially in the case of space dependent parameters or broken translational invariance.
In th with the space depe space-time evolution equations.Using the method we establish the right relation between quantum walk and "TDSE with a space dependent imaginary diffusion coefficient" or "the advection equation with space dependent der to examine its practicality.Moreover our findings provide a clue to find more general space dependent formalisms such as solution method of TDSE with space dependent resolution by quantum walk/QCA.

Appendix Appendix A (Continuum Limit of Quantum Walk with Constant Parameters)
Here we briefly review the way how an evolution equation can be derived as a continuum (zero wave number) limit of a quantum walk with constant parameters.This derivation technique is also used when 4-point-averaging is introduced (Appendix C).
In the method described here, the time evolution generator is expanded with respect to wavenumber k around k = 0.This treatment is essentially the same described in other literatures [4,6].
In the latter, the effective mass was given from the dispersion relation cosω(k) = cosθcosk (though their model is different from ours and their θ corresponds to our π 2   ).
Note that the derivation in this appendix is ath not m em uous function atically rigorous, we rather provide an outline of the derivation needed to explain or interpret the background and results of our numerical experiments.
We regard the wavefunction as a contin

 
x  in spa , when the shape of the wavefunction varies slowly ce as compared with the grid spacing of the quantum walk.We show the continuous space-time evolution equation thus introduced.
First we consider the continuum limit of the classical ra tin ndom walk, classical counterpart of the quantum walk.It is well known that by central limit theorem, the conuum limit of the classical random walk is a diffusion equation (If the left-right balance of the walk is broken, it leads to an advection-diffusion equation with an advective term) First, we review how this equation can be derived.We assume that the transition probability matrix P is translationally invariant.Namely P commutes with ne- grid shift (to the negative directio operation matrix Ŝ .Below is a simplest example of classical random w lk o a n) w yclic lattice of size N or periodic bo ith probabilities of both left and right walks being the equivalent value 1/2.
Here we assume c undary condition.
In treating translationally invariant is usual technique to diagonalize discrete system, it both Ŝ and P si- multaneously using Z-transformation.
P s is the diagonal element of the diagonalized here P corresponding to the eigenvalue s of Ŝ .
Next we extend the problem from discrete time to continuous time and assume this leads to the f llowing continuous time evolution equation Therefore in the real space representation by replacing we obtain a diffusion e n quatio   E, F being Now we assume the 2 × 2 matrix form of Then we regard the square root of as 1-walk evolution matrix.
The logarithm of U(s) is obtained by dec into the scalar part and the traceless part as follows.
As has eigenvalues +1 and −1, we can obtain Then using onsi ore concrete form for ( , D).
    exp cos sin cos sin sin cos and therefore In order to investigate the continuum ehavior, we expand limit (or long range limit) b in a Taylor series with respect to k of ik s e  around k = 0 and we have arccos sin cos sin sin arccos sin cos 1 s Here we use the Taylor expansion of arccos, Two cases ( 0   ) are particularly important, and from now on we restrict our arguments to these cases.
arccos sin cos The equation of motion for wavenumber representation is π and 0 2 for Thus we obtain the TDSE with potential term   for 0   and advection equation.
Here we drop higher-order terms.cribe the physical meaning of Now we des briefly.
For TDSE-type ( 0 It is plausible that the eigenvector corresponds to the wave function slowly varying in space,  This can be seen as follows (shown by the article [6]).
Therefore in a real space representation without the constant phase rotation π  in Equation (A.33), 2 we have Finally we comment about unit system.By comparing the continuum limit evolution equation for the TDSEtype quantum walk with  and b ere briefly review the factorizati of 2-grid transtionally invariant unitary banded matrix Factorization of ma [14].
We will show      we let the 2 × 2 unitary matrix of a constant parameter quantum walk t, es at grid let o, p, q, r, s, u, v, w, x, y, z be the wave function valu points as shown in Figure A4.Namely , , , ed values.
By a simple calculation , we can find that this matrix is singular and can have a relation   Now we consider its square root as 1-step evolution matrix.

Figure 6 .Figure 7 .
Figure 6.Comparison of three methods for different scalings (after 100 walks) dash-dotted red: theoretical solution, dotted green: method1 solid blue: met genta: method 3, solid cyan: velocity A(x).When the wave packet locates the region where A(x) is small, the solution of all methods coincide with the theoretical solution with good accuracy.hod 2, dashed ma-

Figure 8 .
Figure 8.Comparison of three methods for different scalings (after 350 walks) dash-dotted red: theoretical solution, dotted green: method 1 solid blue: method 2, dashed magenta: method 3, solid cyan: velocity A(x).When the wave packet locates around the region where A(x) ≈ 1, the differences among these methods become large and only the solution of method 2 coincides with the theoretical solution.mathematicalmethod.And it is indispensable to compare th In the case of quantum walk, we can use basically the sa has translational invariance.(A.8) me technique.the difference is that the quantum walk not an 1-grid translational invariance but a 2of U is obtained by the matrix multiplication of Z-transformation of each factor.Û Ŝ e use Z-transformation of 2 × 2 matrix unit.
we plot the dispersi SE type quantum walk (Equation (A.25)).quadratic to linear and this corresponds to the one-dimensional Dirac equation with a small mass 1

Figure
Figure A1.dispersion relation for TDSE (or advection) type quantum walk.
Figure A3.Band-width reduction of unitary banded matrix.
The only difference between T(s) and U(s) is the eigenfunctions of   Therefore this time, the advection-type quantum walk has an eigenvector and we can assume that the wave function spatially-varying slowly must have only one component out of two