I am playing with matrices which are linear combinations of identity matrix, Pauli spin matrices, $\sigma_x$ and $\sigma_z$ or their tensor products. For example, let the matrix be $H$. So, $H$ could be as follows.
$$H = \alpha \sigma^{\otimes n}_x + \beta \sigma^{\otimes n}_z + \gamma \hat{I}^{\otimes n}$$ where $\alpha, \beta, \gamma \in \mathbb{R}$ and $n \in \mathbb{N}$.
My question is as $n \to \infty$, what will be the asymptotic behavior of $||H||_2$?