Limit Theorems For Quantum Walks Associated with Hadamard Matrices

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.

References in corpus (4)

• Limit distributions of two-dimensional quantum walks
• Wigner formula of rotation matrices and quantum walks
• Quantum walks and orbital states of a Weyl particle
• An analytic solution for one-dimensional quantum walks

Cited by in corpus (1)

• Asymptotic distributions of quantum walks on the line with two entangled coins