Limit theorems for quantum walks associated with Hadamard matrices
Clement Boateng Ampadu · Physical Review A · 2011
We study a one-parameter family of discrete-time quantum walk models on $\mathbb{Z}$ and ${\mathbb{Z}}^{2}$ associated with the Hadamard walk. Weak convergence in the long-time limit of all moments of the walker's pseudovelocity on $\mathbb{Z}$ and ${\mathbb{Z}}^{2}$ is proved. Symmetrization on $\mathbb{Z}$ and ${\mathbb{Z}}^{2}$ is theoretically investigated, leading to the resolution of the Konno-Namiki-Soshi conjecture in the special case of symmetrization of the unbiased Hadamard walk on $\mathbb{Z}$. A necessary condition for the existence of a phenomenon known as localization is given.