Role of the coin in the spectrum of quantum walks

The most elementary quantum walk is characterized by a 2-dimensional unitary coin flip matrix, which can be parameterized by 4 real variables. The influence of the choice of the coin flip matrix on the time evolution operator is analysed in a systematic way. By changing the coin parameters, the dispersion and asymmetry of eigenvalues of the time evolution operator can be tuned in a controlled way. The reduced eigenvectors in coin space are distributed along trajectories on the surface or inside the Bloch sphere, depending on the degeneracy of the spectrum. At certain values of the coin parameters the spectrum of the time evolution operator becomes 2-fold degenerate, but there might exist unique eigenvalues at the top and bottom of each quasi-energy band. The eigenstates corresponding to such eigenvalues are robust against arbitrary temporal variations in the bias parameter of the coin, as long as rest of the parameters remain unchanged.