Stationary amplitudes of quantum walks on the higher-dimensional integer lattice

Abstract

Stationary measures of quantum walks on the one-dimensional integer lattice are well studied. However, the stationary measure for the higher dimensional case has not been clarified. In this paper, we give the stationary amplitude for quantum walks on the d-dimensional integer lattice with a finite support by solving the corresponding eigenvalue problem. As a corollary, we can obtain the stationary measures of the Grover walks. In fact, the amplitude for the stationary measure is an eigenfunction with eigenvalue 1.

10 pages

References in corpus (5)

• Limit distributions of two-dimensional quantum walks
• Relation between two-phase quantum walks and the topological invariant
• Stationary measures of three-state quantum walks on the one-dimensional lattice
• Stationary measures for the three-state Grover walk with one defect in one dimension
• The stationary measure for diagonal quantum walk with one defect