If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.

What is the distribution of the following inner product: $$\Big|\sum_{n=0}^N \mathbf x_n^H\mathbf x_n + \sum_{n=0}^N \mathbf x_n^H\mathbf y_n\Big|^2$$ If the answer is Gamma distribution, what are the parameters of this Gamma distribution? Note that each element in both vectors is complex, random, and independent of the other elements.