We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the distribution of acceptance probabilities of such MO-QFA, which is partly motivated by similar freeness problems for matrix semigroups and other computational models. We show that determining if the acceptance probabilities of all possible input words are unique is undecidable for 32 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial configuration is defined over real algebraic numbers. We utilize properties of the skew field of quaternions, free rotation groups, representations of tuples of rationals as a linear sum of radicals and a reduction of the mixed modification Post’s correspondence problem.
We consider notions of freeness and ambiguity for the acceptance probability of Moore-Crutchfield Measure Once Quantum Finite Automata (MO-QFA). We study the distribution of acceptance probabilities of such MO-QFA, which is partly motivated by similar freeness problems for matrix semigroups and other computational models. We show that determining if the acceptance probabilities of all possible input words are unique is undecidable for 32 state MO-QFA, even when all unitary matrices and the projection matrix are rational and the initial configuration is defined over real algebraic numbers. We utilize properties of the skew field of quaternions, free rotation groups, representations of tuples of rationals as a linear sum of radicals and a reduction of the mixed modification Post’s correspondence problem.
In the past year two different models of quantum finite automata have been proposed. The first model, introduced by Moore and Crutchfield[MC98], makes one measurement on its state at the end of its computation and is called a measure-once quantum finite automata. The second model, introduced by Kondacs and Watrous[KW97], makes a measurement of its state after every transition and is called a measure-many quantum finite automata. In this thesis we investigate the characteristics of the two models. We characterize measure-once quantum finite automata when they are restricted to acceptance with bounded error, and we show that, when they are not so restricted, they can solve the word problem over the free group. We show that they can be simulated by probabilistic finite automata, which implies that they are no more powerful than probabilistic finite automata. We also describe an algorithm that determines if two automata are equivalent. We show that the class of languages accepted by measure-many quantum finite automata is closed under quotient, complement, and inverse homomorphisms. We prove a necessary condition for a language to be accepted by a measure-many automaton with bounded error and we show that certain sets, including piecewise testable sets, can be accepted with bounded error by this automata.
We present an explicit embedding of measure-once quantum finite automata (MO-QFAs) into the Feynman quantum computer model.Given an MO-QFA, we construct the corresponding Feynman machine whose register encodes both the input word and the automaton state, while a cursor evolving under a Hamiltonian implements the sequential application of the automaton transition operators.The construction relies on nested SWITCH mechanisms that select transitions according to the input symbols.We prove that, for every MO-QFA and every input word, measuring the cursor in its final position yields exactly the quantum state reached by the automaton after processing the same input.Consequently, acceptance probabilities are preserved exactly, and every language recognized by an MO-QFA with cutpoint can be recognized within the Feynman framework with the same acceptance behavior.This establishes an explicit connection between discrete-time quantum computation and continuous-time, physically motivated Hamiltonian models.
The 2-way quantum finite automaton introduced by Kondacs and Watrous [Proceedings of the 38th Annual Symposium on Foundations of Computer Science, 1997, IEEE Computer Society, pp. 66--75] can accept nonregular languages with bounded error in polynomial time. If we restrict the head of the automaton to moving classically and to moving only in one direction, the acceptance power of this 1-way quantum finite automaton is reduced to a proper subset of the regular languages. In this paper we study two different models of 1-way quantum finite automata. The first model, termed measure-once quantum finite automata, was introduced by Moore and Crutchfield [Theoret. Comput. Sci., 237 (2000), pp. 275--306], and the second model, termed measure-many quantum finite automata, was introduced by Kondacs and Watrous [Proceedings of the38th Annual Symposium on Foundations of Computer Science, 1997, IEEE Computer Society, pp. 66--75]. We characterize the measure-once model when it is restricted to accepting with bounded error and show that, without that restriction, it can solve the word problem over the free group. We also show that it can be simulated by a probabilistic finite automaton and describe an algorithm that determines if two measure-once automata are equivalent. We prove several closure properties of the classes of languages accepted by measure-many automata, including inverse homomorphisms, and provide a new necessary condition for a language to be accepted by the measure-many model with bounded error. Finally, we show that piecewise testable sets can be accepted with bounded error by a measure-many quantum finite automaton, introducing new construction techniques for quantum automata in the process.
One-way quantum finite automata (1QFA) proposed by Moore and Crutchfield and by Kondacs and Watrous accept only subsets of regular languages with bounded error. In this paper, we develop a new computing model of 1QFA, namely, one-way quantum finite automata together with classical states (1QFAC for short). In this model, a component of classical states is added, and the choice of unitary evolution of quantum states at each step is closely related to the current classical state. 1QFAC can accept all regular languages with no error, and in particular, 1QFAC can accept some languages with essentially less number of states than deterministic finite automata (DFA). The main technical results are as follows. (1) We prove that the set of languages accepted by 1QFAC with bounded error consists precisely of all regular languages. (2) We show that, for any prime number m ≥ 2, there exists a regular language L0(m) whose minimal DFA needs O(m) states and that can not be accepted by the 1QFA proposed by Moore and Crutchfield and by Kondacs and Watrous, but there exists 1QFAC accepting L0(m) with only constant classical states and O(log(m)) quantum basis states. Analogous results for multi-letter automata are also established. (3) By a bilinearization technique we prove that any two 1QFAC A1 and A2 are equivalent if and only if they are (k1n1) 2 + (k2n2) 2 − 1-equivalent, and there exists a polynomial-time O([(k1n1) 2 + (k2n2) 2]4) algorithm for determining their equivalence, where k1 and k2 are the numbers of classical states of A1 and A2, as well as n1 and n2 are the numbers of quantum basis states of A1 andA2, respectively. (4) We show that the minimization problem of 1QFAC is in EXPSPACE. As a corollary of this result, we obtain that the minimization problem of MO-1QFA is also in EXPSPACE. Finally, we draw some conclusions and point out open ∗Corresponding author. E-mail address: issqdw@mail.sysu.edu.cn (D. Qiu). †E-mail address: {paulo.mateus,amilcar.sernadas}@math.ist.utl.pt.
The study of quantum finite automata (QFAs) is one of the possible approaches in exploring quantum computers with finite memory. Despite being one of the most restricted models, Moore-Crutchfield quantum finite automaton (MCQFA) is proven to be exponentially more succinct than classical finite automata models in recognizing certain languages such as MODp={aj∣j≡0modp}, where p is a prime number. In this paper, we present a modified MCQFA algorithm for the language MODp, the operators of which are selected based on the basis gates on the available real quantum computers. As a consequence, we obtain shorter quantum programs using fewer basis gates compared to the implementation of the original algorithm given in the literature.