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In this paper, we consider the time averaged distribution of discrete time quantum walks on the glued trees. In order to analyze the walks on the glued trees, we consider a reduction to the walks on path graphs. Using a spectral analysis of the Jacobi matrices defined by the corresponding random walks on the path graphs, we have a spectral decomposition of the time evolution operator of the quantum walks. We find significant contributions of the eigenvalues,

The discrete time quantum walks (DTQWs) as quantum counterparts of the random walks, which play important roles in various fields, have been attractive research objects in the last decade [

The rest of this paper is organized as follows. The definition of our DTQW is given in Section 2, and the main result of this paper is stated in Section 3. The remaining section (Section 4) is devoted to the proof of our result.

Let _{k}_{k}^{h}^{−1} numbers of vertices with their height =

In this paper, we consider the glued trees, _{k}_{k}_{i}, e_{j}_{i}_{j}_{i}_{j}_{k}

In a quantum search algorithm [_{k}_{2}_{n}_{2}_{n}_{2}_{n}

First of all, we identify all the vertices with their height = _{2}_{n}_{2}_{n}

Recall that the simple random walk on

for the vertices,

if the vertex, _{2}_{n}_{k}

Now, we define corresponding DTQWs, which we call the Szegedy walk [_{2}_{n}_{2}_{n}_{2}_{n}_{2}_{n}^{T}^{T}^{T}A^{(2}^{n}^{)}, on ℋ_{2}_{n}^{(2}^{n}^{)} = ^{(2}^{n}^{)}^{(2}^{n}^{)} with the following coin operator, ^{(2}^{n}^{)}, and shift operator, ^{(2}^{n}^{)}:

where
_{n}

Let

In this paper, we consider the DTQW starting from a vertex, _{2}_{n}_{1} = _{i}_{i}_{2}_{n}

where the expectation takes the choice of the initial chirality state. For the sake of simplicity, we only consider the

In this section, we show our main result on the time averaged distribution,

The time averaged distribution,

Note that the ^{1}(^{2}(_{2}_{n}_{2}_{n}

Indeed, as is shown in Lemma 4.3, the eigenvalues and the eigenvectors of the time evolution operator, ^{(2}^{n}^{)}, are given by that of _{2}_{n}^{(2}^{n}^{)}. Unfortunately, as is shown in Lemma 4.2, we cannot obtain all of the eigenvalues of _{2}_{n}^{(2}^{n}^{)}.

In order to prove Theorem 3.1, we have the following result for the eigenspace of the Jacobi matrices at first:

(The determinantal formula for a symmetric Jacobi matrix). _{2}_{n}

_{i}_{i}_{2}_{n}, i.e.,_{2}_{n}_{2}_{n}

_{k}

The eigenvector corresponding to the eigenvalue, λ, with

The eigenvector corresponding to the eigenvalue, λ, with

We should remark that every eigenvalue of _{2}_{n}

Let:

Note that by exchanging rows and columns, we have det(_{k}_{k}

By expanding det(_{2}_{n}_{2}_{n}

On the other hand, repeating the expansion of the determinants, we obtain:

and:

Therefore, we have:

For the second equality, we use the following relation:

This completes the first half of the proof.

We can easily check that:

and:

for

From the condition

Similarly, from the condition

These conditions imply that the vectors described in the lemma are the corresponding eigenvectors.

Now, we apply Lemma 4.1 with parameters _{1} = · · · = _{n}_{0} = ^{2}, _{2} = · · · = _{n}_{−2} = _{n}_{−1} = _{2}_{n}_{2}_{n}

where

Remark that for

where _{k}

Using these facts, we can calculate:

Therefore, the eigenvalues of _{2}_{n}_{n}_{−1}) _{n}_{−2}) = 0.

For the

with det(_{1}) = _{0}) = 1. This implies:

On the other hand, for the _{k}_{k}

In this case, we obtain:

with _{−1}(_{2}_{n}

_{2}_{n}_{λ}_{λ}_{λ}, is the following:

The last part of Lemma 4.2, _{λ}||^{2}, is calculated as follows:

Here, we us
_{−1}(

The eigen space of the time evolution operator, ^{(2}^{n}^{)}, is described by that of _{2}_{n}

_{k}_{2}_{n}_{λ}_{k}_{1} = 1 _{2}_{n}_{k}_{k}^{(2}^{n}^{)}

_{λ}_{λ}/||_{λ}|| and for k

Now, we estimate the distribution

Using the spectral decomposition

with
^{(2}^{n}^{)} are nondegenerate (this comes from nondegenerateness of _{2}_{n}

for

As a consequence, we have the desired lower bound.

Taking suitable limits, the remaining parts of Theorem 3.1 are obtained by this lower bound.

The eigenvectors, _{±k}

Yusuke Ide was supported by the Grant-in-Aid for Young Scientists (B) of the Japan Society for the Promotion of Science (grant no. 23740093). Norio Konno was supported by the Grant-in-Aid for Scientific Research (C) of the Japan Society for the Promotion of Science (grant no. 24540116). Etsuo Segawa was supported by the Grant-in-Aid for Young Scientists (B) of the Japan Society for the Promotion of Science (grant no. 25800088). Xin-Ping Xu was supported by the National Natural Science Foundation of China under project 11205110.

The authors declare no conflicts of interest.

A glued tree, _{2}(6), consists of two 2-ary trees with height = 3,
_{2}(6), is reduced to the path graph, _{6}.