In quantum computing, quantum finite automata (QFA) or quantum state machines are a quantum analog of probabilistic automata or a Markov decision process. They provide a mathematical abstraction of real-world quantum computers. Several types of automata may be defined, including measure-once and measure-many automata. Quantum finite automata can also be understood as the quantization of subshifts of finite type, or as a quantization of Markov chains. QFAs are, in turn, special cases of geometric finite automata or topological finite automata.

The automata work by receiving a finite-length string ${\displaystyle \sigma =(\sigma _{0},\sigma _{1},\cdots ,\sigma _{k})}$ of letters ${\displaystyle \sigma _{i}}$ from a finite alphabet ${\displaystyle \Sigma }$, and assigning to each such string a probability ${\displaystyle \operatorname {Pr} (\sigma )}$ indicating the probability of the automaton being in an accept state; that is, indicating whether the automaton accepted or rejected the string.

The languages accepted by QFAs are not the regular languages of deterministic finite automata, nor are they the stochastic languages of probabilistic finite automata. Study of these quantum languages remains an active area of research.

There is a simple, intuitive way of understanding quantum finite automata. One begins with a graph-theoretic interpretation of deterministic finite automata (DFA). A DFA can be represented as a directed graph, with states as nodes in the graph, and arrows representing state transitions. Each arrow is labelled with a possible input symbol, so that, given a specific state and an input symbol, the arrow points at the next state. One way of representing such a graph is by means of a set of adjacency matrices, with one matrix for each input symbol. In this case, the list of possible DFA states is written as a column vector. For a given input symbol, the adjacency matrix indicates how any given state (row in the state vector) will transition to the next state; a state transition is given by matrix multiplication.

One needs a distinct adjacency matrix for each possible input symbol, since each input symbol can result in a different transition. The entries in the adjacency matrix must be zero's and one's. For any given column in the matrix, only one entry can be non-zero: this is the entry that indicates the next (unique) state transition. Similarly, the state of the system is a column vector, in which only one entry is non-zero: this entry corresponds to the current state of the system. Let ${\displaystyle \Sigma }$ denote the set of input symbols. For a given input symbol ${\displaystyle \alpha \in \Sigma }$, write ${\displaystyle U_{\alpha }}$ as the adjacency matrix that describes the evolution of the DFA to its next state. The set ${\displaystyle \{U_{\alpha }|\alpha \in \Sigma \}}$ then completely describes the state transition function of the DFA. Let Q represent the set of possible states of the DFA. If there are N states in Q, then each matrix ${\displaystyle U_{\alpha }}$ is N by N-dimensional. The initial state ${\displaystyle q_{0}\in Q}$ corresponds to a column vector with a one in the q₀'th row. A general state q is then a column vector with a one in the q'th row. By abuse of notation, let q₀ and q also denote these two vectors. Then, after reading input symbols ${\displaystyle \alpha \beta \gamma \cdots }$ from the input tape, the state of the DFA will be given by ${\displaystyle q=\cdots U_{\gamma }U_{\beta }U_{\alpha }q_{0}.}$ The state transitions are given by ordinary matrix multiplication (that is, multiply q₀ by ${\displaystyle U_{\alpha }}$, etc.); the order of application is 'reversed' only because we follow the standard notation of linear algebra.

The above description of a DFA, in terms of linear operators and vectors, almost begs for generalization, by replacing the state-vector q by some general vector, and the matrices ${\displaystyle \{U_{\alpha }\}}$ by some general operators. This is essentially what a QFA does: it replaces q by a unit vector, and the ${\displaystyle \{U_{\alpha }\}}$ by unitary matrices. Other, similar generalizations also become obvious: the vector q can be some distribution on a manifold; the set of transition matrices become automorphisms of the manifold; this defines a topological finite automaton. Similarly, the matrices could be taken as automorphisms of a homogeneous space; this defines a geometric finite automaton.

Before moving on to the formal description of a QFA, there are two noteworthy generalizations that should be mentioned and understood. The first is the non-deterministic finite automaton (NFA). In this case, the vector q is replaced by a vector that can have more than one entry that is non-zero. Such a vector then represents an element of the power set of Q; it’s just an indicator function on Q. Likewise, the state transition matrices ${\displaystyle \{U_{\alpha }\}}$ are defined in such a way that a given column can have several non-zero entries in it. Equivalently, the multiply-add operations performed during component-wise matrix multiplication should be replaced by Boolean and-or operations, that is, so that one is working with a ring of characteristic 2.

A well-known theorem states that, for each DFA, there is an equivalent NFA, and vice versa. This implies that the set of languages that can be recognized by DFA's and NFA's are the same; these are the regular languages. In the generalization to QFAs, the set of recognized languages will be different. Describing that set is one of the outstanding research problems in QFA theory.