One-way Quantum Finite Automata Research Articles

An Enhanced Photonic Quantum Finite Automaton

In a recent paper we have described an optical implementation of a measure-once one-way quantum finite automaton recognizing a well-known family of unary periodic languages, accepting words not in the language with a given error probability. To process input words, the automaton exploits the degree of polarization of single photons and, to reduce the acceptance error probability, a technique of confidence amplification using the photon counts is implemented. In this paper, we show that the performance of this automaton may be further improved by using strategies that suitably consider both the orthogonal output polarizations of the photon. In our analysis, we also take into account how detector dark counts may affect the performance of the automaton.

Watson–Crick quantum finite automata

One-way quantum finite automata are reversible in nature, which greatly reduces its accepting property. In fact, the set of languages accepted by one-way quantum finite automata is a proper subset of regular languages. In this paper, we replace the tape head of one-way quantum finite automata with DNA double strand and name the model Watson–Crick quantum finite automata. The non-injective complementarity relation of Watson–Crick automata introduces non-determinism in the quantum model. We show that this introduction of non-determinism increases the computational power of one-way quantum finite automata significantly. Watson–Crick quantum finite automata can accept all regular languages and also accepts some languages which are not accepted by any multi-head deterministic finite automata. Exploiting the superposition property of quantum finite automata, we show that Watson–Crick quantum finite automata accept the language L = {ww|w ∈ {a, b}*}.

Quantum ω-Automata over Infinite Words and Their Relationships

Inspired by the results of finite automata working on infinite words, we studied the quantum ω-automata with Buchi, Muller, Rabin and Streett acceptance condition. Quantum finite automata play a pivotal part in quantum information and computational theory. Investigation of the power of quantum finite automata over infinite words is a natural goal. We have investigated the classes of quantum ω-automata from two aspects: the language recognition and their closure properties. It has been shown that quantum Muller automaton is more dominant than quantum Buchi automaton. Furthermore, we have demonstrated the languages recognized by one-way quantum finite automata with different quantum acceptance conditions. Finally, we have proved the closure properties of quantum ω-automata.

On the power of two-way multihead quantum finite automata

This paper introduces a variant of two-way quantum finite automata named two-way multihead quantum finite automata. A two-way quantum finite automaton is more powerful than classical two-way finite automata. However, the generalizations of two-way quantum finite automata have not been defined so far as compared to one-way quantum finite automata model. We have investigated the newly introduced automata from two aspects: the language recognition capability and its comparison with classical and quantum counterparts. It has been proved that a language which cannot be recognized by any one-way and multi-letter quantum finite automata can be recognized by two-way quantum finite automata. Further, it has been shown that a language which cannot be recognized by two-way quantum finite automata can be recognized by two-way multihead quantum finite automata with two heads. Furthermore, it has been investigated that quantum variant of two-way deterministic multihead finite automata takes less number of heads to recognize a language containing of all words whose length is a prime number.

Power of the interactive proof systems with verifiers modeled by semi-quantum two-way finite automata

Interactive proof systems (IP) are very powerful – languages they can accept form exactly PSPACE. They represent also one of the very fundamental concepts of theoretical computing and a model of computation by interactions. One of the key players in IP is verifier. In the original model of IP whose power is that of PSPACE, the only restriction on verifiers is that they work in randomized polynomial time. Because of such key importance of IP, it is of large interest to find out how powerful will IP be when verifiers are more restricted. So far this was explored for the case that verifiers are two-way probabilistic finite automata (Dwork and Stockmeyer, 1990) and one-way quantum finite automata as well as two-way quantum finite automata (Nishimura and Yamakami, 2009). IP in which verifiers use public randomization is called Arthur–Merlin proof systems (AM). AM with verifiers modeled by Turing Machines augmented with a fixed-size quantum register (qAM) were studied also by Yakaryilmaz (2012). He proved, for example, that an NP-complete language Lknapsack, representing the 0–1 knapsack problem, can be recognized by a qAM whose verifier is a two-way finite automaton working on quantum mixed states using superoperators.In this paper we explore the power of AM for the case that verifiers are two-way finite automata with quantum and classical states (2QCFA) – introduced by Ambainis and Watrous in 2002 – and the communications are classical. It is of interest to consider AM with such “semi-quantum” verifiers because they use only limited quantum resources. Our main result is that such Quantum Arthur–Merlin proof systems (QAM(2QCFA)) with polynomial expected running time are more powerful than the models in which the verifiers are two-way probabilistic finite automata (AM(2PFA)) with polynomial expected running time. Moreover, we prove that there is a language which can be recognized by an exponential expected running time QAM(2QCFA), but cannot be recognized by any AM(2PFA), and that the NP-complete language Lknapsack can also be recognized by a QAM(2QCFA) working only on quantum pure states using unitary operators.

Constant-space quantum interactive proofs against multiple provers

We present upper and lower bounds of the computational complexity of the two-way communication model of multiple-prover quantum interactive proof systems whose verifiers are limited to measure-many two-way quantum finite automata. We prove that (i) the languages recognized by those multiple-prover systems running in expected polynomial time are exactly the ones in NEXP, the nondeterministic exponential-time complexity class, (ii) if we further require verifiers to be one-way quantum finite automata, then their associated proof systems recognize context-free languages but not beyond languages in NE, the nondeterministic linear exponential-time complexity class, and moreover, (iii) when no time bound is imposed, the proof systems become as powerful as Turing machines. The first two results answer affirmatively an open question, posed by Nishimura and Yamakami [J. Comput. System Sci. 75 (2009) 255–269], of whether multiple-prover quantum interactive proof systems are more powerful than single-prover ones. Our proofs are simple and intuitive, although they heavily rely on an earlier result on multiple-prover classical interactive proof systems of Feige and Shamir [J. Comput. System Sci. 44 (1992) 259–271].

Superiority of one-way and realtime quantum machines

In automata theory, quantum computation has been widely examined for finite state machines, known as quantum finite automata (QFAs), and less attention has been given to QFAs augmented with counters or stacks. In this paper, we focus on such generalizations of QFAs where the input head operates in one-way or realtime mode, and present some new results regarding their superiority over their classical counterparts. Our first result is about the nondeterministic acceptance mode: Each quantum model architecturally intermediate between realtime finite state automaton and one-way pushdown automaton (one-way finite automaton, realtime and one-way finite automata with one-counter, and realtime pushdown automaton) is superior to its classical counterpart. The second and third results are about bounded error language recognition: for any k > 0, QFAs with k blind counters outperform their deterministic counterparts; and, a one-way QFA with a single head recognizes an infinite family of languages, which can be recognized by one-way probabilistic finite automata with at least two heads. Lastly, we compare the nondeterminictic and deterministic acceptance modes for classical finite automata with k blind counter(s), and we show that for any k > 0, the nondeterministic models outperform the deterministic ones.

Characterizations of one-way general quantum finite automata

Generally, unitary transformations limit the computational power of quantum finite automata (QFA). In this paper, we study a generalized model named one-way general quantum finite automata (1gQFA), in which each symbol in the input alphabet induces a trace-preserving quantum operation, instead of a unitary transformation. Two different kinds of 1gQFA will be studied: measure-once one-way general quantum finite automata (MO-1gQFA) where a measurement deciding to accept or reject is performed at the end of a computation, and measure-many one-way general quantum finite automata (MM-1gQFA) where a similar measurement is performed after each trace-preserving quantum operation on reading each input symbol. We characterize the measure-once model from three aspects: the closure property, the language recognition power, and the equivalence problem. We prove that MO-1gQFA recognize, with bounded error, precisely the set of all regular languages. Our results imply that some models of quantum finite automata proposed in the literature, which were expected to be more powerful, still cannot recognize non-regular languages. We prove that MM-1gQFA also recognize only regular languages with bounded error. Thus, MM-1gQFA and MO-1gQFA have the same language recognition power, in sharp contrast with traditional MO-1QFA and MM-1QFA, the former being strictly less powerful than the latter. Finally, we present a necessary and sufficient condition for two MM-1gQFA to be equivalent.

Multi-letter quantum finite automata: decidability of the equivalence and minimization of states

Multi-letter quantum finite automata (QFAs) can be thought of quantum variants of the one-way multi-head finite automata (Hromkovic, Acta Informatica 19:377–384, 1983). It has been shown that this new one-way QFAs (multi-letter QFAs) can accept with no error some regular languages, for example (a + b)*b, that are not acceptable by QFAs of Moore and Crutchfield (Theor Comput Sci 237:275–306, 2000) as well as Kondacs and Watrous (66–75, 1997; Observe that 1-letter QFAs are exactly measure-once QFAs (MO-1QFAs) of Moore and Crutchfield (Theor Comput Sci 237:275–306, 2000)). In this paper, we study the decidability of the equivalence and minimization problems of multi-letter QFAs. Three new results presented in this paper are the following ones: (1) Given a k 1-letter QFA $${{\mathcal A}_1}$$ and a k 2-letter QFA $${{\mathcal A}_2}$$ over the same input alphabet Σ, they are equivalent if and only if they are (n 2 m k-1−m k-1 + k)-equivalent, where m = |Σ| is the cardinality of Σ, k = max(k 1,k 2), and n = n 1 + n 2, with n 1 and n 2 being numbers of states of $${{\mathcal A}_{1}}$$ and $${{\mathcal A}_{2}}$$, respectively. When k = 1, this result implies the decidability of equivalence of measure-once QFAs (Moore and Crutchfield in Theor Comput Sci 237:275–306, 2000). (It is worth mentioning that our technical method is essentially different from the previous ones used in the literature.) (2) A polynomial-time O(m 2k-1 n 8 + km k n 6) algorithm is designed to determine the equivalence of any two multi-letter QFAs (see Theorems 2 and 3; Observe that if a brute force algorithm to determine equivalence would be used, as suggested by the decidability outcome of the point (1), the worst case time complexity would be exponential). Observe also that time complexity is expressed here in terms of the number of states of the multi-letter QFAs and k can be seen as a constant. (3) It is shown that the states minimization problem of multi-letter QFAs is solvable in EXPSPACE. This implies also that the state minimization problem of MO-1QFAs (see Moore and Crutchfield in Theor Comput Sci 237:275–306, 2000, page 304, Problem 5), an open problem stated in that paper, is also solvable in EXPSPACE.

Languages recognized by nondeterministic quantum finite automata

The nondeterministic quantum finite automaton (NQFA) is the only known case where a one-way quantum finite automaton (QFA) model has been shown to be strictly superior in terms of language recognition power to its probabilistic counterpart. We give a characterization of the class of languages recognized by NQFAs, demonstrating that it is equal to the class of exclusive stochastic languages. We also characterize the class of languages that are recognized necessarily by two-sided error by QFAs. It is shown that these classes remain the same when the QFAs used in their definitions are replaced by several different model variants that have appeared in the literature. We prove several closure properties of the related classes. The ramifications of these results about classical and quantum sublogarithmic space complexity classes are examined.

Hierarchy and equivalence of multi-letter quantum finite automata

Multi-letter quantum finite automata (QFAs) are a new one-way QFA model proposed recently by Belovs, Rosmanis, and Smotrovs [A. Belovs, A. Rosmanis, J. Smotrovs, Multi-letter reversible and quantum finite automata, in: Proceedings of the 13th International Conference on Developments in Language Theory, DLT’2007, Harrachov, Czech Republic, in: Lecture Notes in Computer Science, vol. 4588, Springer, Berlin, 2007, pp. 60–71], and they showed that multi-letter QFAs can accept with no error some regular languages ( ( a + b ) ∗ b ) that are unacceptable by the one-way QFAs. In this paper, we continue to study multi-letter QFAs. We mainly focus on two issues: (1) we show that ( k + 1 ) -letter QFAs are computationally more powerful than k -letter QFAs, that is, ( k + 1 ) -letter QFAs can accept some regular languages that are unacceptable by any k -letter QFA. A comparison with the one-way QFAs is made by some examples; (2) we prove that a k 1 -letter QFA A 1 and another k 2 -letter QFA A 2 are equivalent, if and only if, they are ( n 1 + n 2 ) 4 + k − 1 -equivalent, and the time complexity of determining the equivalence of two multi-letter QFAs using this method is O ( n 12 + k 2 n 4 + k n 8 ) , where n 1 and n 2 are the numbers of states of A 1 and A 2 , respectively, and k = max ( k 1 , k 2 ) . Some other issues are addressed for further consideration.

GOLOMB RULERS AND DIFFERENCE SETS FOR SUCCINCT QUANTUM AUTOMATA

Given a function p : N → [0,1] of period n, we study the minimal size (number of states) of a one-way quantum finite automaton (Iqfa) inducing the stochastic event ap + b, for real constants a>0, b≥0, a+b≤1. First of all, we relate the estimation of the minimal size to the problem of finding a minimal difference cover for a suitable subset of Zn. Then, by observing that the cardinality of a difference cover Δ for a set A ⊆ Zn, must satisfy [Formula: see text], we investigate the class of sets A admitting difference covers of cardinality exactly [Formula: see text]. We relate this problem with the efficient construction of Golomb rulers and difference sets. We design an algorithm which outputs each of the Golomb rulers (if any) of a given set in pseudo-polynomial time. As a consequence, we obtain an efficient algorithm that construct minimal difference covers for a non-trivial class of sets. Moreover, by using projective geometry arguments, we give an algorithm that, for any n=q2+q+1 with q prime power, constructs difference sets for Zn in quadratic time.

On the Size of One-way Quantum Finite Automata with Periodic Behaviors

We show that, for any stochastic event p of period n, there exists a measure-once one-way quantum finite automaton (1qfa) with at most 26n + 25 states inducing the event ap + b, for constants a > 0, b ≥ 0, satisfying a + b < 1. This fact is proved by designing an algorithm which constructs the desired lqfa in polynomial time. As a consequence, we get that any periodic language of period n can be accepted with isolated cut point by a lqfa with no more than 26n+26 states. Our results give added evidence of the strength of measure-once 1qfa's with respect to classical automata.