On hybrid models of quantum finite automata

Abstract

In the literature, there exist several quantum finite automata (QFA) models with both quantum and classical states. These models are of particular interest,as they show praiseworthy advantages over the fully quantum models in some nontrivial aspects. This paper characterizes these models in a uniform framework by proposing a general hybrid model consisting of a quantum component and a classical one which can interact with each other. The existing hybrid QFA can be naturally regarded as the general model with specific communication patterns (classical-quantum, quantum-classical, and two-way, respectively). We further clarify the relationship between these hybrid QFA and some other quantum models. In particular, it is shown that hybrid QFA can be simulated exactly by QFA with quantum operations, which in turn has a close relationship with two early proposed models: {\it ancialla QFA} and {\it quantum sequential machines}.Comment: The paper has been revised such that it is more concise and has a better structure. All comments are welcom

Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game

“…Variants of quantum finite automata, placing emphasis on hybrid models, were presented in [29] by Li and Feng, where they obtained interesting theoretical results demonstrating the advantages of these models. The use of such finite state machines for the representation of quantum games could, perhaps, constitute an alternative to classical automata, particularly in view of some encouraging results regarding their power and expressiveness (see [30][31][32]).…”

Quantum Markov chains: Description of hybrid systems, decidability of equivalence, and model checking linear-time properties

“…[3,5,21,30,31,29]). In fact, these automata in the mentioned references as hybrid systems have been described in a uniform way by the authors [14]. When viewing HQA as language acceptors, we show their language equivalence problem is decidable in polynomial time by transforming this problem to the equivalence problem of probabilistic automata.…”

“…Quantum data are represented by states of the quantum component and classical control flows are state evolutions of the classical component. Also note that the quantum Turing machines defined in [26] and quantum finite automata studied in [14] are all hybrid systems. In addition, as shown in [8], quantum cryptographic protocols such as BB84 protocol can be described by hqMC.…”

“…(i) It is obvious that a probabilistic automaton is a degenerate C-HQA in which M (s, t) = p s,t I for all s, t ∈ S with (p s,t ) s,t∈S being a stochastic matrix, and µ 0 (s) = p s ρ for some density operator ρ and s ∈ S with (p s ) s∈S being a probabilistic distribution. (ii) Note that we characterized three models of quantum finite automata in the framework of hybrid systems in [14]; all of them can be regarded as a concrete implementation of the HQA model defined in this paper. For example, CL-1QFA [5] and 1QCFA [30] are instances of C-HQA, and 1QFAC [21] are instances of Q-HQA.…”

“…Specially, we will show that this model is very suited to describe hybrid systems that consists of a quantum component and a classical one. Indeed, hybrid systems are often encountered in quantum computing and quantum information, varying from quantum Turing machines [26] and quantum finite automata [14,22] to quantum programs [24] and quantum protocols such as BB84. Quantum engineering systems developed in the future will most probably have a classical human-interactive interface and a quantum processor, and thus they will be hybrid models.…”