Hadamard Matrix Method
Given that one basis in a Hilbert space is the standard basis, then all bases which are unbiased with respect to this basis can be represented by the columns of a complex Hadamard matrix multiplied by a normalization factor. For d = 3 these matrices would have the form
The problem of finding a set of k+1 mutually unbiased bases therefore corresponds to finding k mutually unbiased complex Hadamard matrices.
An example of a one parameter family of Hadamard matrices in a 4 dimensional Hilbert space is
Read more about this topic: Mutually Unbiased Bases, Methods For Finding Mutually Unbiased Bases
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