  For certain real quadratic fields $K$ with sufficiently small discriminant we
produce explicit unit generators for specific ray class fields of $K$ using a
numerical method that arose in the study of complete sets of equiangular lines
in $\mathbb{C}^d$ (known in quantum information as symmetric informationally
complete measurements or SICs). The construction in low dimensions suggests a
general recipe for producing unit generators in infinite towers of ray class
fields above arbitrary real quadratic $K$, and we summarise this in a
conjecture. There are indications [19,20] that the logarithms of these
canonical units are related to the values of $L$-functions associated to the
extensions, following the programme laid out in the Stark Conjectures.
