The automata work by accepting a finite-length string of letters from a finite alphabet , and assigning to each such string a probability indicating the probability of the automaton being in an accept state; that is, indicating whether the automaton accepted or rejected the string.
As with an ordinary finite automaton, the quantum automaton is considered to have possible internal states, represented in this case by an -state qubit . More precisely, the -state qubit is an element of -dimensional complex projective space, carrying an inner product that is the Fubini-Study metric.
The state transitions, transition matrixes or de Bruijn graphs are represented by a collection of unitary matrixes , with one unitary matrix for each letter . That is, given an input letter , the unitary matrix describes the transition of the automaton from its current state to its next state :
Thus, the triple form a quantum semiautomaton.
The probability of the state machine accepting a given finite input string is given by
Here, the vector is understood to represent the initial state of the automaton, that is, the state the automaton was in before it stated accepting the string input. The empty string 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 on reverses the order of the letters in the string , it is not uncommon for QFA's to be defined using a right action on the Hermitian transpose states, simply in order to keep the order of the letters the same.
A regular language is accepted with probability by a quantum finite automaton, if, for all sentences in the language, (and a given, fixed initial state ), one has
with the complex numbers normalized so that
The unitary transition matrices are
Taking to be the accept state, the projection matrix is
As should be readily apparent, if the initial state is the pure state or , then the result of running the machine will be exactly identical to the classical deterministic finite state machine. In particular, there is a language accepted by this automaton with probability one, for these initial states, and it is identical to the regular language for the classical DFA, and is given by the regular expression:
The non-classical behaviour occurs if both and are non-zero. More subtle behaviour occurs when the matrices and are not so simple; see, for example, the de Rham curve as an example of a quantum finite state machine acting on the set of all possible finite binary strings.
In the literature, these orthogonal subspaces are usually formulated in terms of the set of orthogonal basis vectors for the Hilbert space . This set of basis vectors is divided up into subsets and , such that
is the linear span of the basis vectors in the accept set. The reject space is defined analogously, and the remaining space is designated the non-halting subspace. There are three projection matrices, , and , each projecting to the respective subspace:
and so on. The parsing of the input sring proceeds as follows. Consider the automaton to be in a state . After reading an input letter , the automaton will be in the state
At this point, a measurement is performed on the state , using the projection operators , at which time its wave-function collapses into one of the three subspaces or or . The probability of collapse is given by
for the "accept" subspace, and analogously for the other two spaces.
If the wave function has collapsed to either the "accept" or "reject" subspaces, then further processing halts. Otherwise, processing continues, with the next letter read from the input, and applied to what must be an eigenstate of . Processng continues until the whole string is read, or the machine halts. Often, additional symbols and $ are adjoined to the alphabet, to act as the left and right end-markers for the string.
In the literature, the meaure-many automaton is often denoted by the tuple . Here, , , and are as defined above. The initial state is denoted by . The unitary transformations are denoted by the map ,
The quantum automaton differs from the topological automaton in that, instead of having a binary result (is the iterated point in, or not in, the final set?), one has a probability. The quantum probability is the (square of) the initial state projected onto some final state P; that is . But this probability amplitude is just a very simple function of the distance between the point and the point in , 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 is generalized to some metric space, and the probability measure is replaced by a simple function of the metric on that space.
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