Abstract Learning finite automata (termed as model learning) has become an important field in machine learning and has been useful realistic applications. Quantum finite automata (QFA) are simple models of quantum computers with finite memory. Due to their simplicity, QFA have well physical realizability, but one-way QFA still have essential advantages over classical finite automata with regard to state complexity (two-way QFA are more powerful than classical finite automata in computation ability as well). As a different problem in quantum learning theory and quantum machine learning, in this paper, our purpose is to initiate the study of learning QFA with queries (naturally it may be termed as quantum model learning), and the main results are regarding learning two basic one-way QFA (1QFA): (1) we propose a learning algorithm for measure-once 1QFA (MO-1QFA) with query complexity of polynomial time and (2) we propose a learning algorithm for measure-many 1QFA (MM-1QFA) with query complexity of polynomial time, as well.
Abstract Learning finite automata (termed as model learning) has become an important field in machine learning and has been useful realistic applications. Quantum finite automata (QFA) are simple models of quantum computers with finite memory. Due to their simplicity, QFA have well physical realizability, but one-way QFA still have essential advantages over classical finite automata with regard to state complexity (two-way QFA are more powerful than classical finite automata in computation ability as well). As a different problem in quantum learning theory and quantum machine learning, in this paper, our purpose is to initiate the study of learning QFA with queries (naturally it may be termed as quantum model learning), and the main results are regarding learning two basic one-way QFA (1QFA): (1) we propose a learning algorithm for measure-once 1QFA (MO-1QFA) with query complexity of polynomial time and (2) we propose a learning algorithm for measure-many 1QFA (MM-1QFA) with query complexity of polynomial time, as well.
One-way quantum finite automata together with classical states (1QFAC) proposed in [Journal of Computer and System Sciences 81(2) (2015) 359--375] is a new one-way quantum finite automata (1QFA) model that integrates quantum finite automata (QFA) and deterministic finite automata (DFA). This model uses classical states to control the evolution and measurement of quantum states. As a quantum-classical hybrid model, 1QFAC recognize all regular languages. It was shown that the state complexity of 1QFAC for some languages is essentially superior to that of DFA and other 1QFA. In this paper, our goal is to clarify state complexity problems for 1QFAC. We obtain the following results: (1) We optimize the bound given by Qiu et al. that characterizes the relationship between quantum basis state number and classical state number of 1QFAC as well as the state number of its corresponding minimal DFA for recognizing any given regular language. (2) We give an upper bound showing that how many classical states are needed if the quantum basis states of 1QFAC are reduced without changing its recognition ability. (3) We give a lower bound of the classical state number of 1QFAC for recognizing any given regular language, and the lower bound is exact if the given language is finite. (4) We show that 1QFAC are exponentially more succinct than DFA and probabilistic finite automata (PFA) for recognizing some regular languages that can not be recognized by measure-once 1QFA (MO-1QFA), measure-many 1QFA (MM-1QFA) or multi-letter 1QFA. (5) We reveal essential relationships between 1QFAC, MO-1QFA and multi-letter 1QFA, and induce a result regarding a quantitative relationship between the state number of multi-letter 1QFA and DFA.
We show that there are quantum devices that accept all regular languages and that are exponentially more concise than deterministic finite automata (DFA). For this purpose, we introduce a new computing model of {\it one-way quantum finite automata} (1QFA), namely, {\it one-way quantum finite automata together with classical states} (1QFAC), which extends naturally both measure-only 1QFA and DFA and whose state complexity is upper-bounded by both. The original contributions of the paper are the following. First, we show that the set of languages accepted by 1QFAC with bounded error consists precisely of all regular languages. Second, we prove that 1QFAC are at most exponentially more concise than DFA. Third, we show that the previous bound is tight for families of regular languages that are not recognized by measure-once (RMO), measure-many (RMM) and multi-letter 1QFA. % More concretely we exhibit regular languages L0(m) for m prime such that: (i) L0(m) cannot be recognized by measure-once, measure-many and multi-letter 1QFA; (ii) the minimal DFA that accepts L0(m) has O(m) states; (iii) there is a 1QFAC with constant classical states and O(log(m)) quantum basis that accepts L0(m). Fourth, we give a polynomial-time algorithm for determining whether any two 1QFAC are equivalent. Finally, we show that state minimization of 1QFAC is decidable within EXPSPACE. We conclude the paper by posing some open problems.
We study 1-way quantum finite automata (QFAs). First, we compare them with their classical counterparts. We show that, if an automaton is required to give the correct answer with a large probability (greater than 7/9), then any 1-way QFAs can be simulated by a 1-way reversible automaton. However, quantum automata giving the correct answer with smaller probabilities are more powerful than reversible automata. Second, we show that 1-way QFAs can be very space-efficient. We construct a 1-way QFA that is exponentially smaller than any equivalent classical (even randomized) finite automaton. We think that this construction may be useful for design of other space-efficient quantum algorithms. Third, we consider several generalizations of 1-way QFAs. Here, our goal is to find a model which is more powerful than 1-way QFAs keeping the quantum part as simple as possible.
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.
In this paper, we introduce 1-way and 2-way quantum finite state automata (1qfa's and 2qfa's), which are the quantum analogues of deterministic, nondeterministic and probabilistic 1-way and 2-way finite state automata. We prove the following facts regarding 2qfa's. 1. For any /spl epsiv/>0, there is a 2qfa M which recognizes the non-regular language L={a/sup m/b/sup m/|m/spl ges/1} with (one-sided) error bounded by E, and which halts in linear time. Specifically, M accepts any string in L with probability 1 and rejects any string not in L with probability at least 1-/spl epsiv/. 2. For every regular language L, there is a reversible (and hence quantum) 2-way finite state automaton which recognizes L and which runs in linear time. In fact, it is possible to define 2qfar's which recognize the non-context-free language {a/sup m/b/sup m/c/sup m/|m/spl ges/1}, based on the same technique used for 1. Consequently, the class of languages recognized by linear time, bounded error 2qfa's properly includes the regular languages. Since it is known that 2-way deterministic, nondeterministic and polynomial expected time, bounded error probabilistic finite automata can recognize only regular languages, it follows that 2qfa's are strictly more powerful than these "classical" models. In the case of 1-way automata, the situation is reversed. We prove that the class of languages recognizable by bounded error 1qfa's is properly contained in the class of regular languages.
Multi-letter {\it quantum finite automata} (QFAs) were a quantum variant of classical {\it one-way multi-head finite automata} (J. Hromkovič, Acta Informatica 19 (1983) 377-384), and it has been shown that this new one-way QFAs (multi-letter QFAs) can accept with no error some regular languages (a+b)∗b that are unacceptable by the previous one-way QFAs. In this paper, we study the decidability of the equivalence of multi-letter QFAs, and the main technical contributions are as follows: (1) We show that any two automata, a k1-letter QFA A1 and a k2-letter QFA A2, over the same input alphabet Σ are equivalent if and only if they are (n2mk−1−mk−1+k)-equivalent, where m=∣Σ∣ is the cardinality of Σ, k=max(k1,k2), and n=n1+n2, with n1 and n2 being the numbers of states of A1 and A2, respectively. When k=1, we obtain the decidability of equivalence of measure-once QFAs in the literature. It is worth mentioning that our technical method is essentially different from that for the decidability of the case of single input alphabet (i.e., m=1). (2) However, if we determine the equivalence of multi-letter QFAs by checking all strings of length not more than $ n^2m^{k-1}-m^{k-1}+k$, then the worst time complexity is exponential, i.e., O(n6mn2mk−1−mk−1+2k−1). Therefore, we design a polynomial-time O(m2k−1n8+kmkn6) algorithm for determining the equivalence of any two multi-letter QFAs. Here, the time complexity is concerning the number of states in the multi-letter QFAs, and k is thought of as a constant.