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Mathematical analysis on the existence of eigenvalues is vital, as itcorresponds to the occurrence of localization, an exceptionally importantproperty of quantum walks. Previous studies have demonstrated that eigenvalueanalysis utilizing the transfer matrix proves beneficial for spaceinhomogeneous three-state quantum walks with a specific class of coin matrices,including Grover matrices. In this research, we turn our attention to thetransfer matrix of three-state quantum walks with a general coin matrix.Building upon previous research methodologies, we dive deeper intoinvestigating the properties of the transfer matrix and employ numericalanalysis to derive eigenvalues for models that were previously unanalyzable.

Eigenvalue analysis of three-state quantum walks with general coin matrices