The discrete-time quaternionic quantum walk on a graph. (English) Zbl 1333.81210
Summary: Recently, the quaternionic quantum walk was formulated by the first author as a generalization of discrete-time quantum walks. We deal with the right eigenvalue problem of quaternionic matrices in order to study spectra of the transition matrix of a quaternionic quantum walk. The way to obtain all the right eigenvalues of a quaternionic matrix is given. From the unitary condition on the transition matrix of a quaternionic quantum walk, we deduce some remarkable properties of it. Our main results determine all the right eigenvalues of the quaternionic quantum walk by using those of the corresponding weighted matrix. In addition, we give some examples of quaternionic quantum walks and their right eigenvalues.
MSC:
| 81S25 | Quantum stochastic calculus |
| 60G50 | Sums of independent random variables; random walks |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 60F05 | Central limit and other weak theorems |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 11R52 | Quaternion and other division algebras: arithmetic, zeta functions |
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