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Mutually unbiased bases that can be cyclically generated by a single unitaryoperator are of special interest, since they can be readily implemented inpractice. We show that, for a system of qubits, finding such a generator can becast as the problem of finding a symmetric matrix over the field $\mathbb{F}_2$equipped with an irreducible characteristic polynomial of a given Fibonacciindex. The entanglement structure of the resulting complete sets is determinedby two additive matrices of the same size.

Structure of the sets of mutually unbiased bases with cyclic symmetry