Stationary amplitudes of quantum walks on the higher-dimensional integer lattice
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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.
KeywordsDiscrete time quantum walk Stationary amplitude Higher-dimensional integer lattice Stationary measure
This work is partially supported by the Grant-in-Aid for Scientific Research (Challenging Exploratory Research) of Japan Society for the Promotion of Science (Grant No. 15K13443).
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