  We consider the task of deciding whether an unknown qubit state falls in a
prescribed neighborhood of a reference state. We assume that several copies of
the unknown state are given and apply a unitary operation pairwise on them
combined with a post-selection scheme conditioned on the measurement result
obtained on one of the qubits of the pair. The resulting transformation is a
deterministic, nonlinear, chaotic map in the Hilbert space. We derive a class
of these transformations capable of orthogonalizing nonorthogonal qubit states
after a few iterations. These nonlinear maps orthogonalize states which
correspond to the two different convergence regions of the nonlinear map. Based
on the analysis of the border (the so-called Julia set) between the two regions
of convergence, we show that it is always possible to find a map capable of
deciding whether an unknown state is within a neighborhood of fixed radius
around a desired quantum state. We analyze which one- and two-qubit operations
would physically realize the scheme. It is possible to find a single two-qubit
unitary gate for each map or, alternatively, a universal special two-qubit gate
together with single-qubit gates in order to carry out the task. We note that
it is enough to have a single physical realization of the required gates due to
the iterative nature of the scheme.
