3-state Grover and Fourier quantum walks on C_N have finite period only for N=3 (T_3=6 and 12), via a cyclotomic field method that gives a necessary condition on coin operators.
R., Quantum state revivals in quantum walks on c ycles, Results in Physics, 4, 189–197 (2014)
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Periodicity for the 3-state quantum walk on cycles
3-state Grover and Fourier quantum walks on C_N have finite period only for N=3 (T_3=6 and 12), via a cyclotomic field method that gives a necessary condition on coin operators.