Eigenvalues of Two-State Quantum Walks Induced by the Hadamard Walk

Area of Science:

• Quantum mechanics
• Condensed matter physics
• Quantum information theory

Background:

• The localization of discrete-time quantum walks is intrinsically linked to the existence of their eigenvalues.
• Previous studies have focused on homogeneous quantum walks, leaving inhomogeneous cases less explored.

Purpose of the Study:

• To determine the eigenvalue distributions for one-dimensional space-inhomogeneous quantum walks.
• To elucidate the relationship between these eigenvalue distributions and quantum walk localization.
• To investigate the influence of characteristic parameters on eigenvalue distributions.

Main Methods:

• Utilized the splitted generating function method (SGF method) for eigenvalue analysis.
• Employed numerical simulations to explore parameter dependencies.
• Focused on one-dimensional, space-inhomogeneous quantum walk models.

Main Results:

• Successfully revealed the distributions of eigenvalues for space-inhomogeneous quantum walks.
• Demonstrated a clear dependence of these distributions on characteristic parameters.
• Provided the first comprehensive analysis of eigenvalue distributions in this context.

Conclusions:

• The eigenvalue distributions of space-inhomogeneous quantum walks are now characterized.
• Understanding these distributions offers new insights into quantum walk localization phenomena.
• The SGF method proves effective for analyzing complex quantum walk dynamics.