 The accept state of the automaton is given by an formula_27 projection matrix formula_36, so that, given a formula_20-dimensional quantum state formula_22, the probability of formula_22 being in the accept state is

The probability of the state machine accepting a given finite input string formula_1 is given by

Here, the vector formula_22 is understood to represent the initial state of the automaton, that is, the state the automaton was in before it started accepting the string input.The empty string formula_44 is understood to be just the unit matrix, so that

is just the probability of the initial state being an accepted state.Because the left-action of formula_7 on formula_22 reverses the order of the letters in the string formula_48, it is not uncommon for QFAs to be defined using a right action on the Hermitian transpose states, simply in order to keep the order of the letters the same.