 But this probability amplitude is just a very simple function of the distance between the point formula_109 and the point formula_110 in formula_17, under the distance metric given by the Fubini–Study metric.To recap, the quantum probability of a language being accepted can be interpreted as a metric, with the probability of accept being unity, if the metric distance between the initial and final states is zero, and otherwise the probability of accept is less than one, if the metric distance is non-zero.Thus, it follows that the quantum finite automaton is just a special case of a geometric automaton or a metric automaton, where formula_17 is generalized to some metric space, and the probability measure is replaced by a simple function of the metric on that space.