With the [help from Stephen Luttrell in stackexchange][1],  I can give an intuitive expression of quantum walk, as shown in the Mathematica codes below.

    Needs["Notation`"]
    Notation[ParsedBoxWrapper[
    RowBox[{
    RowBox[{
    RowBox[{
    RowBox[{"\[LeftBracketingBar]", "spin_"}], "\[RightAngleBracket]"}], 
       "\[CircleTimes]", 
    RowBox[{"\[LeftBracketingBar]", "index_"}]}], 
     "\[RightAngleBracket]"}]] \[DoubleLongLeftRightArrow] 
    ParsedBoxWrapper[
    RowBox[{"state", "[", 
    RowBox[{"spin_", ",", "index_"}], "]"}]]]
    H[state[spin : ("\[UpArrow]" | "\[DownArrow]"), index_]] := 
    1/Sqrt[2] state["\[UpArrow]", index] + 
    1/Sqrt[2] If[spin == "\[UpArrow]", 1, -1] state["\[DownArrow]", 
     index];
    S[state[spin : ("\[UpArrow]" | "\[DownArrow]"), index_]] := 
    state[spin, index + If[spin == "\[UpArrow]", 1, -1]];
    H[u_ + v_] := H[u] + H[v];
    S[u_ + v_] := S[u] + S[v];
    H[u_?(FreeQ[#, state] &) v_] := u H[v];
    S[u_?(FreeQ[#, state] &) v_] := u S[v];
    state["\[DownArrow]", 0]
    H[%];
    S[%];
    H[%];
    S[%];
    H[%];
    S[%];
    H[%];
    S[%] // Simplify

The result (the fourth step in quantum walk) is
![enter image description here][2]
Based on the previous codes, now I have a new problem :

First, I need to merge the  elements with the same    `\[LeftBracketingBar]index\[RightAngleBracket]`, 
  and rearrange the elements in an increasing order of   `\[LeftBracketingBar]index\[RightAngleBracket]`.
For example, in the fourth step, ![enter image description here][3] should be merged and rearranged as![enter image description here][4]

Second, I need to summarize the coefficient  and calculate the intensity for 
each `\[LeftBracketingBar]index\[RightAngleBracket]`. The intensity is square of the coefficient of `\[LeftBracketingBar]"\[DownArrow]"\[RightAngleBracket]`, puls the   square of the coefficient of `\[LeftBracketingBar]"\[UpArrow]"\[RightAngleBracket]`. For example, in the fourth step, the intensity is  

    1/16 (1; 9 + 1; 1 + 1; 1 + 1; 1)=(1/16; 5/8; 1/8; 1/8; 1/16)
My final goal is to make the following table, which is the famous quantum walk table. The x axis is the position, and the y axis is the step number.
![enter image description here][5]



My question is how to summarize the coefficient in Notation using Mathematica codes? 
Any help or suggestion is highly appreciated.



  [1]: https://mathematica.stackexchange.com/questions/58362/how-to-put-the-tensor-product-of-two-operators-onto-two-variables
  [2]: https://i.sstatic.net/zSAEO.jpg
  [3]: https://i.sstatic.net/BPi2w.jpg
  [4]: https://i.sstatic.net/9fj2A.jpg
  [5]: https://i.sstatic.net/xrVGC.jpg