# What is this equation for coin operator is trying to do in this quantum walk for Non-regular graph? This coin operator is called Fourier coin

I am reading the following paper: Discrete-time quantum walk on complex networks for community detection by Kanae Mukai
We define the Coin operator $$C$$ by: $$C=C_1\otimes C_2....C_n$$ , We define coin operator for Node $$i, C_i:H_i\to H_i$$ is given by:

$$C_i^F|i\to j_1\rangle|i\to j_2\rangle.......|i\to j_k\rangle=(|i\to j_1\rangle|i\to j_2\rangle.......|i \to j_k\rangle)1/\sqrt(k_i) \begin{pmatrix} 1 & 1 & 1 & ... & 1\\ 1 & e^{\iota\theta/k_i} & e^{2\iota\theta/k_i} & ... & e^{(k_i-1)\iota\theta/k_i}\\ 1 & e^{2\iota\theta/k_i} & e^{4\iota\theta/k_i} & ... & e^{2(k_i-1)\iota\theta/k_i}\\ . & . & . & . & .\\. & . & . & . & .\\. & . & . & . & .\\ 1 & e^{(k_i-1)\iota\theta/k_i} & e^{2(k_i-1)\iota\theta/k_i} & ... & e^{(k_i-1)(k_i-1)\iota\theta/k_i}\end{pmatrix}$$
Here $$k_i$$ is the degree of $$i^{th}$$ node and $$\theta=2\pi$$. The author called this coin as Fourier Coin. And this $$i\to j$$ implies that Node $$i$$ is going to hope to adjacent Node $$j$$. Now, What is going on with this equation?

• Does anyone have any idea, I really need to know this quickly? – Binshumesh sachan Nov 21 '20 at 13:16

The operator Fourier Coin is $$k$$-point Discrete Fourier Transform (DFT) of node $$i$$. The matrix representation of a general $$N$$-point DFT can be found here.