Limit Theorems For Quantum Walks Associated with Hadamard Matrices

TL;DR

This paper establishes limit theorems for quantum walks related to Hadamard matrices, demonstrating weak convergence of moments and addressing symmetrization effects and localization phenomena in these models.

Contribution

It provides the first proof of weak convergence for moments of Hadamard quantum walks and resolves the Konno-Namiki-Soshi conjecture in a specific symmetrized case.

Findings

Proved weak convergence of all moments of the walker's pseudo-velocity.

Resolved the Konno-Namiki-Soshi conjecture for symmetrized unbiased Hadamard walks.

Identified a necessary condition for localization phenomena.

Abstract

We study a one-parameter family of discrete-time quantum walk models on the line and in the xy-plane associated with the Hadamard walk. Weak convergence in the long-time limit of all moments of the walker's pseudo-velocity on the line and in the xy-plane is proved. Symmetrization on the line and in the xy-plane 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 the line . A necessary condition for the existence of a phenomenon known as localization is given.