Abstract
We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow at time n. From this expression, we compute the scattering on the surface of and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.
1. Introduction
There is no doubt that a study on scattering theory is one of the most interesting topics of the Schrödinger equation. Recently, it has been revealed that the scatterings of some fundamental stationary Schrödinger equations on the real line with not only delta potentials [,,] but also continuous potential [] can be recovered by discrete-time quantum walks. These induced quantum walks are given by the following setting: the non-trivial quantum coins are assigned to some vertices in a finite region on the one-dimensional lattice as the impurities and the free-quantum coins are assigned at the other vertices. The initial state is given so that a quantum walker inflows into the perturbed region at every time step. It is shown that the scattering matrix of the quantum walk on the one-dimensional lattice can be explicitly described by using a path counting in [] and this path counting method can be described by a discrete analogue of the Feynmann path integral []. There are some studies for the scattering theory of quantum walks under slightly general settings and related topics [,,,,,,].
Such a setting is the special setting of [,] in that the regions where a quantum walker moves freely coincide with tails in [,], and the perturbed region can be regarded as a finite and connected graph in [,]. The properties of not only the scattering on the surface of the internal graph but also the stationary state in the internal graph for the Szegedy walk are characterized by [] with a constant inflow from the tails.
By [], this quantum walk converges to a stationary state. Therefore, let be the stationary state of the quantum walk on . The perturbed region is and we assign the quantum coin
to each vertex in . The inflow into the perturbed region at time n is expressed by . In this paper, we compute (1) the scattering on the surface of the perturbed region in the one-dimensional lattice; (2) the energy of the quantum walk. Here, the energy of quantum walk is defined by
This is the quantity that quantum walkers accumulate to the perturbed region in the long time limit. We obtain a necessary and sufficient condition for the perfect transmitting, and also obtain the energy. As a consequence of our result on the energy, we observe a discontinuity of the energy with respect to the frequency of the inflow. Moreover, our result implies that the condition for is equivalent to the condition for the perfect transmitting. Then, we obtain that the situation of the perfect transmitting not only releases quantum walker to the opposite outside but also accumulates quantum walkers in the perturbed region. Note that since this quantum walk can be converted to a quantum walk with absorption walls, the problem is reduced to analysis on a finite matrix , which is obtained by picking up from the total unitary time evolution operator with respect to the perturbed region . See [] for a precise spectral results on .
This paper is organized as follows. In Section 2, we explain the setting of this model and give some related works. In Section 3, an explicit expression for the stationary state is computed using the Chebyshev polynomials. From this expression, we obtain the transmitting and reflecting rates and a necessary and sufficient condition for the perfect transmitting. We also give the energy in the perturbed region. In Section 4, we estimate the asymptotics of the energy to see the discontinuity with respect to the incident inflow.
2. The Setting of our Quantum Walk
The total Hilbert space is denoted by . Here A is the set of arcs of one-dimensional lattice whose elements are labeled by , where and represents the arcs “from to x“, and “from to x”, respectively. We assign a unitary matrix to each so-called local quantum coin
Putting , and , , we define the following matrix valued weights associated with the motion from x to left and right by
respectively. Then, the time evolution operator on is described by
for any . Its equivalent expression on is described by
for any . We call and the transmitting amplitudes, and and the reflection amplitudes at x, respectively. If we put and , then the primitive form of QW in [] is reproduced. Remark that U and are unitarily equivalent such that letting be
then we have . The free quantum walk is the quantum walk where all local quantum coins are described by the identity matrix, i.e.,
Then, the walker runs through one-dimensional lattices without any reflections in the free case.
In this paper, we set “impurities” on
in the free quantum walk on one-dimensional lattice; that is,
We consider the initial state as follows.
where . Note that this initial state belongs to no longer category. The region is obtained a time dependent inflow from the negative outside. On the other hand, if a quantum walker goes out side of , it never come back again to . We can regard such a quantum walker as an outflow from . Roughly speaking, in the long time limit, the inflow and outflow are balanced and obtain the stationary state with some modification. Indeed, the following statement holds.
Proposition 1
([]).
- 1.
- This quantum walk converges to a stationary state in the following meaning:
- 2.
- This stationary state is a generalized eigenfunction satisfying
Relation to an absorption problem
Let the reflection amplitude at time n be with . We can see that is rewritten by using as follows:
The first term is the amplitude that the inflow at time n cannot penetrate into ; the m-th term is the amplitude that the inflow at time penetrates into and escapes from 0 side at time n. Therefore, each term corresponds to the “absorption” amplitude to with the absorption walls and M with the initial state . Then
Remark 1.
The reflection amplitude coincides with the generating function of the absorption amplitude to with respect to time n while the transmitting amplitude coincides with the generating function of the absorption amplitude to M with respect to time n.
Put and which are the absorption/ first hitting probabilities at positions and M, respectively, starting from . From the above observation, for example, we can express the m-th moments of the absorption/hitting times to and M as follows:
Relation to Scattering of quantum walk
The stationary state is a generalized eigenfunction of U in . The scattering matrix naturally appears in (see []). In the time independent scattering theory, the inflow can be considered as the incident “plane wave“, and the impurity causes the scattered wave by transmissions and reflections. Thus, we can see the transmission coefficient and the reflection coefficient in for . For studies of a general theory of scattering, we also mention the recent work by Tiedra de Aldecoa [].
3. Computation of Stationary State
3.1. Preliminary
Recall that and represent the standard basis of ; that is, and . Let be a boundary operator such that for any . Here, the adjoint is described by
We put the principal submatrix of U with respect to the impurities by . The matrix form of with the computational basis is expressed by the following matrix:
We express the element of by
Putting , we have
Then, putting , we have
From [], . Then, the stationary state restricted to satisfies
About the uniqueness of this solution is ensured by the following Lemma since it includes the existence of the inverse of .
Lemma 1.
Let be the above with . Then .
Proof.
Let be an eigenvector of eigenvalue . Then
Here, for the inequality, we used the fact that is the projection operator onto
while for the final equality, we used the fact that is the identity operator on . If the equality in (8) holds, then holds. Then, we have the eigenequation by taking to both sides of the original eigenequation . However, there are no eigenvectors having finite supports in a position independent quantum walk on with since its spectrum is described by only a continuous spectrum in general. Thus, . □
Now, let us solve this Equation (7). The matrix representation of with the permutation of the labeling such that for any to (5) is
Then, the Equation (7) is expressed by
Here, we changed the way of blockwise of and we put . Putting
we have
where for any . The inverse matrix of exists since . Then, we have
where
Here . For the boundaries, there exists such that
By (10) and (11), satisfies
which is equivalent to
Now, the problem is reduced to considering the n-th power of T because the eigenvector is expressed by . Since T is a just matrix, we can prepare the following lemma.
Lemma 2.
Let A be a 2-dimensional matrix denoted by
- 1.
- and for some ϵ case. Let . Then
- 2.
- Otherwise. Let for . Then
Remark 2.
The condition “ and ” is equivalent to the non-diagonalizability of A.
Remark 3.
For case, the condition of 1. is reduced to
Remark 4.
For case, the variable of the Chebyshev polynomial in 2. is reduced to
Moreover, if , the Chebyshev polynomial is described by ,
Here, in RHS are the roots of the quadratic equation
with .
3.2. Transmitting and Reflecting Rates
Let us divide the unit circle in the complex plain as follows:
By the unitarity of and using the Chebyshev recursion; , we insert (1) and (2) in Lemma 2 into (10), and we have an explicit expression for the stationary state as follows.
Theorem 1.
Let the stationary state restricted to be and . Then we have
for , where and , . Here and
Since the transmitting and reflecting rates are computed by
we obtain explicit expressions for them as follows.
Corollary 1.
Assume . For any , we have
Note that the unitarity of the time evolution can be confirmed by . By Corollary 1, we can find a necessary and sufficient conditions for the perfect transmitting; that is, .
Corollary 2.
Assume . Let with some real value k. Then the perfect transmitting happens if and only if
On the other hand, the perfect reflection never occurs.
Remark that if , then the perfect transmitting never happens.
3.3. Energy in the Perturbed Region
Taking the square modulus to in Theorem 1, the relative probability at position can be computed as follows.
Proposition 2.
Assume . Then, the relative probability is described by
Proof.
Let us consider the case for . Using the property of the Chebyshev polynomial, we have and . It holds that
Since , we have
Then, we have
□
Then, we can see how much quantum walkers accumulate in the perturbed region by
We call it the energy of quantum walk. The dependency of the energy on is symmetric on the unit circle in the complex plain.
Corollary 3.
Let be the above and assume . Then we have
In particular, is continuous at every and
Proof.
Using the properties of the Chebyshev polynomial for example, , , we have
Then, we have
Then, we have
Here, we used (19) in the last equality.
If , then by directly computation taking summation of (17) over , we obtain the conclusion. Let us see is continuous at . We put and . Remark that implies . In the following, we consider case. The Taylor expansion of around is
The reason for obtaining the expansion until order is
around . Note that . Then
around . Then inserting all of them into (18), we obtain
□
4. Asymptotics of Energy
If , then by Corollary 3, it is immediately obtained that
Let us consider the case of as follows. Note that
where (), while () such that and . To observe the asymptotics of for , we rewrite as follows: -4.6cm0cm
From now on, let us consider the asymptotics of for large M. We summarize our results on the asymptotics of in Table 1. In the following, we regard as a function of , M; that is because can be expressed by and consider the asymptotics for large M.

Table 1.
Asymptotics of the energy of : , .
4.1.
Let us see that
Note that . Then by (21), we have
By (22), if , then . To connect it to the limit for the case of described by (20) continuously, we consider and simultaneously, so that . Let us see that
Noting that , for and , we have
Therefore, if we design the parameter so that
then the energy of continuously closes to that of in the sufficient large system size M.
4.2.
In this paper, since we determine satisfying , we set . Remark that for any because is invariant under this deformation.
By (21), if , we have
for sufficiently large M, which implies that
if is fixed. Then, we conclude that if is fixed for . On the other hand, if we design so that the condition of the perfect transmitting is satisfied; , (see Corollary 1) and choose ℓ which is very close to 0 or M, then . Note that if , which means , then the coefficient of the upper bound in (26) diverges.
Then, from now on, let us consider the following three cases having a magnitude relation between and M;
(i) ; (ii) ; (iii) .
- Case (i):Let us start to evaluate RHS of (21). SinceThis is consistent with (20).
- Case (ii):Under this condition, the parameter lives around 0 or if M is large. Since we consider , we can evaluate by , or for large M. We define if and if . Because by the assumption, we have . Therefore, we put with and . Then up to the value , let us seeNote that if , then and , and so on. Inserting them into (21), we haveOn the other hand, if , since and , by (21), we have
- Case (iii):The “” part in (21) is estimated by because . Then, we have
- (a)
- (b)
- Since , we evaluate byThen, there exists a natural number m such that . Note that is also sufficiently small. Then, the natural number m must be if and if . Putting , we haveTherefore, holds. Then, (29) implies
We summarize the above statements in the following theorem by setting , as a special but natural design of the parameters.
Theorem 2.
Let us set so that
with the parameters and . If or with fixed M, then . On the other hand, if we take and fix , then we have
5. Conclusions
We considered the quantum walk on the line with the perturbed region ; that is, an non-trivial quantum coin is assigned at the perturbed region and the free quantum coin is assigned at the other region. We set an initial state so that free quantum walkers are inputted at each time step to the perturbed region. A closed form of the stationary state of this dynamical system was obtained and we computed the energy of the quantum walk in the perturbed region. This energy represents how quantum walker feels “comfortable“ in the perturbed region. We showed that the “feeling” of quantum walk depends on the frequency of the initial state. We can divide the region of the frequency into three parts to classify the asymptotics of the energy for large M; , , . The region coincides with the continuous spectrum of the quantum walk with []. We showed that quantum walkers prefer to the initial state whose frequency corresponds to the continuous spectrum in the infinite system. More precisely, the energy of the quantum walk in the perturbed region is estimated by if , while one is estimated by if and almost all pseudo momentum gives -energy, but some momentum gives if (Theorem 2). Such an initial state exactly exists but it is quite rare from the view point of the Lebesgue measure. The most comfortable initial state for quantum walkers has the frequency whose pseudo momentum lives in some neighborhood of the boundary and accomplishes the perfect transmitting. If the momentum of the initial state exceeds the boundary from the internal region , then the energy is immediately reduced to . It suggests that the control of the frequency of the initial state to give the maximal energy in the perturbed region is quite sensitive from the view point of an implementation.
The spectrum of the boundary for produces the two singular points of the density function of the Konno limit distribution and is characterized by the Airy functions. In [], details of the spectrum behavior around is discussed. Indeed, a kind of “speciality“ also appears as the non-diagonalizability of T when in our work (Lemma 2). Note that the infinite system does not have any edges, which means every node is “impurity”, while our quantum walker feels the edges of the impurities; nodes 0 and M. Therefore, to see the effect of such a finiteness on the behavior of the quantum walker comparing with the infinite system, computing how a quantum walker is distributed in the perturbed region is interesting which may be possible from the explicit expression of the stationary state in Theorem 1. Moreover, to consider the escaping time from the perturbed region seems to be useful to estimate the finesse as the interferometer motivated by quantum walk and it would be possible to extract some information from (3) and (4). This remains one of the interesting problems for the future.
Author Contributions
Conceptualization, E.S.; Formal analysis, E.S., H.M., T.K., K.H. and N.K. All authors have read and agreed to the published version of the manuscript.
Funding
HM was supported by the grant-in-aid for young scientists No. 16K17630, JSPS. ES acknowledges financial supports from the Grant-in-Aid of Scientific Research (C) No. JP19K03616, Japan Society for the Promotion of Science and Research Origin for Dressed Photon.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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