  We study Anderson localization in a discrete-time quantum map dynamics in one
dimension with nearest-neighbor hopping strength $\theta$ and quasienergies
located on the unit circle. We demonstrate that strong disorder in a local
phase field yields a uniform spectrum gaplessly occupying the entire unit
circle. The resulting eigenstates are exponentially localized. Remarkably this
Anderson localization is universal as all eigenstates have one and the same
localization length $L_{loc}$. We present an exact theory for the calculation
of the localization length as a function of the hopping,
$1/L_\text{loc}=\left|\ln\left(|\sin(\theta)|\right)\right|$, that is tunable
between zero and infinity by variation of the hopping $\theta$.
