Quantum Automata Research Articles

Dominant Strategies of Quantum Games on Quantum Periodic Automata

Game theory and its quantum extension apply in numerous fields that affect people’s social, political, and economical life. Physical limits imposed by the current technology used in computing architectures (e.g., circuit size) give rise to the need for novel mechanisms, such as quantum inspired computation. Elements from quantum computation and mechanics combined with game-theoretic aspects of computing could open new pathways towards the future technological era. This paper associates dominant strategies of repeated quantum games with quantum automata that recognize infinite periodic inputs. As a reference, we used the PQ-PENNY quantum game where the quantum strategy outplays the choice of pure or mixed strategy with probability 1 and therefore the associated quantum automaton accepts with probability 1. We also propose a novel game played on the evolution of an automaton, where players’ actions and strategies are also associated with periodic quantum automata.

Nonlinear optical Galton board: Thermalization and continuous limit.

The nonlinear optical Galton board (NLOGB), a quantum walk like (but nonlinear) discrete time quantum automaton, is shown to admit a complex evolution leading to long time thermalized states. The continuous limit of the Galton board is derived and shown to be a nonlinear Dirac equation (NLDE). The (Galerkin-truncated) NLDE evolution is shown to thermalize toward states qualitatively similar to those of the NLOGB. The NLDE conserved quantities are derived and used to construct a stochastic differential equation converging to grand canonical distributions that are shown to reproduce the (microcanonical) NLDE thermalized statistics. Both the NLOGB and the Galerkin-truncated NLDE are thus demonstrated to exhibit spontaneous thermalization.

Free Quantum Field Theory from Quantum Cellular Automata

After leading to a new axiomatic derivation of quantum theory, the new informational paradigm is entering the domain of quantum field theory, suggesting a quantum automata framework that can be regarded as an extension of quantum field theory to including an hypothetical Planck scale, and with the usual quantum field theory recovered in the relativistic limit of small wave-vectors. Being derived from simple principles (linearity, unitarity, locality, homogeneity, isotropy, and minimality of dimension), the automata theory is quantum ab-initio, and does not assume Lorentz covariance and mechanical notions. Being discrete it can describe localized states and measurements (unmanageable by quantum field theory), solving all the issues plaguing field theory originated from the continuum. These features make the theory an ideal framework for quantum gravity, with relativistic covariance and space-time emergent solely from the interactions, and not assumed a priori. The paper presents a synthetic derivation of the automata theory, showing how from the principles lead to a description in terms of a quantum automaton over a Cayley graph of a group. Restricting to Abelian groups we show how the automata recover the Weyl, Dirac and Maxwell dynamics in the relativistic limit. We conclude with some new routes about the more general scenario of non-Abelian Cayley graphs.

ON THE DECIDABILITY OF THE INTERSECTION PROBLEM FOR QUANTUM AUTOMATA AND CONTEXT-FREE LANGUAGES

We consider the so-called measure once finite quantum automata model introduced by Moore and Crutchfield in 2000. We show that given a language recognized by such a device and a linear context-free language, it is decidable whether or not they have a nonempty intersection. This extends a result of Blondel et al. which can be interpreted as solving the problem with the free monoid in place of the family of linear context-free languages.

Path-integral solution of the one-dimensional Dirac quantum cellular automaton

Quantum cellular automata have been recently considered as a fundamental approach to quantum field theory, resorting to a precise automaton, linear in the field, for the Dirac equation in one dimension. In such linear case a quantum automaton is isomorphic to a quantum walk, and a convenient formulation can be given in terms of transition matrices, leading to a new kind of discrete path integral that we solve analytically in terms of Jacobi polynomials versus the arbitrary mass parameter.

Model-Checking Linear-Time Properties of Quantum Systems

We define a formal framework for reasoning about linear-time properties of quantum systems in which quantum automata are employed in the modeling of systems and certain (closed) subspaces of state Hilbert spaces are used as the atomic propositions about the behavior of systems. We provide an algorithm for verifying invariants of quantum automata. Then, an automata-based model-checking technique is generalized for the verification of safety properties recognizable by reversible automata and ω--properties recognizable by reversible Büchi automata.

Algebraic Characterization of the Class of Languages Recognized by Measure Only Quantum Automata

We study a model of one-way quantum automaton where only measurement operations are allowed (MON-1QFA). We give an algebraic characterization of LMO(Σ), showing that the syntactic monoids of the languages in LMO(Σ) are exactly the J-trivial literally idempotent syntactic monoids, where J is the Green's relation determined by two-sided ideals. We also prove that LMO(Σ) coincides with the literal variety of literally idempotent piecewise testable languages. This allows us to prove the existence of a polynomial-time algorithm for deciding whether a regular language belongs to LMO(Σ).

ESTENSIONE DELLA TEORIA QUANTISTICA DI CAMPO A TEORIA DI AUTOMA CELLULARE QUANTISTICO

In the present note I will briefly report recent results on the novel field theory based on the information-theoretical paradigm. This program continues the previous one on the axiomatic derivation of quantum theory from information-theoretical principles, of which I presented a note on January 31st 2008. The passage from the quantum theory of abstract systems to the theory of fields is achieved by adding to the axioms of quantum theory new general principles for the network of interation among a denumerable set of quantum systems. Such principles can be synthesized with the requirement of minimal complexity of the quantum algorithm describing the physical law. From general principles such as homogeneity, isotropy, locality, linearity, and unitarity of the interaction network, along with minimal dimension of the field vector, one derives the free quantum field theory, namely Weyl, Dirac, and Maxwell fields. The principles lead to a description in terms of a quantum cellular automaton on the Cayley graph of a group. The advantages of the new theory are numerous. For example the theory tautologically solves all the problems originating from the continuum, such as the ultraviolet divergencies, the localization problem, and causality violations. The mechanics itself is not assumed, but it is a consequence of the principles, and the theory is quantum ab initio, namely it does not need quantization of a classical field theory. Lorentz covariance is not assumed, but follows itself from the principles, in the relativistic limit of small wavevectors. The notion of reference system is restated in terms of irreducible representation of the quantum automaton. Finally, the new theory, derived from purely mathematical principles–whence with adimensional variables–contains itself the units of measure for distance, time, and mass, which are given in terms of extremal values of the variables, which are in principle experimentally detectable. Also effects with GR flavor emerge from the automaton theory, such as an upper bound for the Dirac particle rest-mass (from the unitariety condition), where the dispersion relation becomes completely flat, in strict analogy with the notion of mini black hole.

Trace monoids with idempotent generators and measure-only quantum automata

In this paper, we analyze a model of 1-way quantum automaton where only measurements are allowed ( MON -1qfa). The automaton works on a compatibility alphabet $$(\Sigma, E)$$ of observables and its probabilistic behavior is a formal series on the free partially commutative monoid $$\hbox{FI}(\Sigma, E)$$ with idempotent generators. We prove some properties of this class of formal series and we apply the results to analyze the class $${\bf LMO}(\Sigma, E)$$ of languages recognized by MON -1qfa's with isolated cut point. In particular, we prove that $${\bf LMO}(\Sigma, E)$$ is a boolean algebra of recognizable languages with finite variation, and that $${\bf LMO}(\Sigma, E)$$ is properly contained in the recognizable languages, with the exception of the trivial case of complete commutativity.

Efficient quantum processing of three-manifold topological invariants

A quantum algorithm for approximating efficiently three–manifold topological in the framework of $SU(2)$ Chern–Simons–Witten (CSW) topological quantum field theory at finite values of the coupling constant $k$ is provided. The model of computation adopted is the $q$-deformed spin network model viewed as a quantum recognizer in the sense of [1], where each basic unitary transition function can be efficiently processed by a standard quantum circuit. This achievement is an extension of the algorithm for approximating polynomial of colored oriented links found in Spin networks, quantum automata and link invariants and An efficient quantum algorithm for colored Jones polynomials. Thus all the significant quantities — partition functions and observables — of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field-theoretic solvability of such theory at finite $k$. The paper is supplemented by a critical overview of the basic conceptual tools underlying the construction of quantum of links and three–manifolds and connections with algorithmic questions that arise in geometry and quantum gravity models are discussed.

Minimization of a quantum automaton: The transducer

A new model of a Quantum Automaton (QA), working with qubits is proposed. The quantum states of the automaton can be pure or mixed and are represented by density operators. This is the appropriated approach to deal with measurements and dechorence. The linearity of a QA and of the partial trace super-operator, combined with the properties of invariant subspaces under unitary transformations, are used to minimize the dimension of the automaton and, consequently, the number of its working qubits. The results here developed are valid wether the state set of the QA is finite or not. There are two main results in this paper: 1) We show that the dimension reduction is possible whenever the unitary transformations, associated to each letter of the input alphabet, obey a set of conditions. 2) We develop an algorithm to find out the equivalent minimal QA and prove that its complexity is polynomial in its dimension and in the size of the input alphabet.

A note on quantum sequential machines

Quantum sequential machines (QSMs) are a quantum version of stochastic sequential machines (SSMs). Recently, we showed that two QSMs M 1 and M 2 with n 1 and n 2 states, respectively, are equivalent iff they are ( n 1 + n 2 ) 2 -equivalent [L.Z. Li, D.W. Qiu, Determination of equivalence between quantum sequential machines, Theoretical Computer Science 358 (2006) 65–74]. However, using this result to check the equivalence is likely to need exponential expected time. In this note, we consider the time complexity of deciding the equivalence between QSMs and related problems. The main results are as follows: (1) We present a polynomial-time algorithm for deciding the equivalence between QSMs, and, if two QSMs are not equivalent, this algorithm will produce an input–output pair with length not more than ( n 1 + n 2 ) 2 . (2) We improve the bound for the equivalence between QSMs from ( n 1 + n 2 ) 2 to n 1 2 + n 2 2 − 1 , by employing Moore and Crutchfield’s method [C. Moore, J.P. Crutchfield, Quantum automata and quantum grammars, Theoretical Computer Science 237 (2000) 275–306. Also quant-ph/9707031, 1997]. In addition, by viewing MO-1QFAs as a special case of QSMs, we briefly discuss the equivalence between MO-1QFAs, where the method used and the result obtained are slightly different from those given by Koshiba [T. Koshiba, Polynomial-time algorithms for the equivalence for one-way quantum finite automata, in: Proceedings of the 12th International Symposium on Algorithms and Computation, ISAAC’2001, Christchurch, New Zealand, in: Lecture Notes in Computer Science, vol. 2223, Springer, Berlin, 2001, pp. 268–278].

Weakly Regular Quantum Grammars and Asynchronous Quantum Automata

In this paper, we define weakly regular quantum grammars (WRQG), regular quantum grammars (RQG), asynchronous quantum automata (AQA) and synchronous quantum automata (SQA). Moreover, we investigate the relationships between quantum languages generated by weakly quantum regular grammars and by asynchronous quantum automata. At the mean time, we discuss the relationships between regular quantum grammars and synchronous quantum automata.

An overview of quantum computation models: quantum automata

Quantum automata, as theoretical models of quantum computers, include quantum finite automata (QFA), quantum sequential machines (QSM), quantum pushdown automata (QPDA), quantum Turing machines (QTM), quantum cellular automata (QCA), and the others, for example, automata theory based on quantum logic (orthomodular lattice-valued automata). In this paper, we try to outline a basic progress in the research on these models, focusing on QFA, QSM, QPDA, QTM, and orthomodular lattice-valued automata. Also, other models closely relative to them are mentioned. In particular, based on the existing results in the literature, we finally address a number of problems to be studied in future.

SOME REMARKS ON QUANTUM AUTOMATA

A uniform mathematical description of quantum automata as quantum finite automata, quantum linear bounded automata, and quantum Turing machines is introduced, and the existence of such automata under certain circumstances is investigated. Such quantum automata are modelled by isometric rather than unitary operators, reflecting the fact that computation is primarily irreversible in contrast to physical phenomena in the microworld. The global conditions determine the local conditions of the operators.

Topological Automata

We give a new, topological definition of automata that extends previous definitions of probabilistic and quantum automata. We then are able to prove in a unified framework that deterministic or non-deterministic probabilistic and quantum automata recognise only regular languages with an isolated threshold.

Quantum geometry and quantum algorithms

Motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature, we provide a quantum algorithm that efficiently approximates the coloured Jones polynomial. The construction is based on the complete solution of the Chern–Simons topological quantum field theory and its connection to Wess–Zumino–Witten conformal field theory. The coloured Jones polynomial is expressed as the expectation value of the evolution of the q-deformed spin-network quantum automaton. A quantum circuit is constructed capable of simulating the automaton and hence of computing such an expectation value. The latter is efficiently approximated using a standard sampling procedure in quantum computation.

Quantum Computation of Universal Link Invariants

In the framework of the spin-network simulator based on the SU q(2) tensor algebra, we implement families of finite state quantum automata capable of accepting the language generated by the braid group, and whose transition amplitudes are coloured Jones polynomials. The automaton calculation of the polynomial of a link L on n strands at any fixed root of unity q is bounded from above by a linear function of the number of crossings of the link, on the one hand, and polynomially bounded in terms of the braid index n, on the other.

Algebraic Results on Quantum Automata

We use tools from the algebraic theory of automata to investigate the class of languages recognized by two models of Quantum Finite Automata (QFA): Brodsky and Pippenger’s end-decisive model, and a new QFA model whose definition is motivated by implementations of quantum computers using nucleo-magnetic resonance (NMR). In particular, we are interested in the new model since nucleo-magnetic resonance was used to construct the most powerful physical quantum machine to date. We give a complete characterization of the languages recognized by the new model and by Boolean combinations of the Brodsky-Pippenger model. Our results show a striking similarity in the class of languages recognized by the end-decisive QFAs and the new model, even though these machines are very different on the surface.

Finite State and Finite Stop Quantum Languages

We propose the concept of finite stop quantum automata (ftqa) based on Hilbert space and compare it with the finite state quantum automata (fsqa) proposed by Moore and Crutchfield (Theoretical Computer Science237(1–2), 2000, 275–306). The languages accepted by fsqa form a proper subset of the languages accepted by ftqa. In addition, the fsqa form an infinite hierarchy of language inclusion with respect to the dimensionality of unitary matrices. We introduce complex-valued acceptance degrees and two types of finite stop quantum automata based on them: the invariant ftqa (icftq) and the variant ftqa (vcftq). The languages accepted by icftq form a proper subset of the languages accepted by vcftq. In addition, the icftq form an infinite hierarchy of language inclusion with respect to the dimensionality of unitary matrices. In this way, we establish two proper inclusion relations \({\cal L}\) (fsqa) ⊂ \({\cal L}\) (ftqa) and \({\cal L}\) (icftq) ⊂ \({\cal L}\) (vcftq), where the symbol \({\cal L}\) means languages, and two infinite language hierarchies \({\cal L}_{n}\) (fsqa) ⊂ \({\cal L}_{n+1}\) (fsqa), \({\cal L}_{n}\) (icftq) \({\cal L}_{n+1}\) (icftq).