Quantum Pushdown Automata

Quantum finite automata, as well as quantum pushdown automata (QPA) were first introduced by C. Moore and J. P. Crutchfield. In this paper we introduce the notion of QPA in a non-equivalent way, including unitarity criteria, by using the definition of quantum finite automata of Kondacs and Watrous. It is established that the unitarity criteria of QPA are not equivalent to the corresponding unitarity criteria of quantum Turing machines. We show that QPA can recognize every regular language. Finally we present some simple languages recognized by QPA, not recognizable by deterministic pushdown automata.Comment: Conference SOFSEM 2000, extended version of the pape

Superiority of one-way and realtime quantum machines and new directions

In automata theory, the quantum computation has been widely examined for finite state machines, known as quantum finite automata (QFAs), and less attention has been given to the QFAs augmented with counters or stacks. Moreover, to our knowledge, there is no result related to QFAs having more than one input head. In this paper, we focus on such generalizations of QFAs whose input head(s) operate(s) in one-way or realtime mode and present many superiority of them to their classical counterparts. Furthermore, we propose some open problems and conjectures in order to investigate the power of quantumness better. We also give some new results on classical computation.Comment: A revised edition with some correction

Quantum counter automata

The question of whether quantum real-time one-counter automata (rtQ1CAs) can outperform their probabilistic counterparts has been open for more than a decade. We provide an affirmative answer to this question, by demonstrating a non-context-free language that can be recognized with perfect soundness by a rtQ1CA. This is the first demonstration of the superiority of a quantum model to the corresponding classical one in the real-time case with an error bound less than 1. We also introduce a generalization of the rtQ1CA, the quantum one-way one-counter automaton (1Q1CA), and show that they too are superior to the corresponding family of probabilistic machines. For this purpose, we provide general definitions of these models that reflect the modern approach to the definition of quantum finite automata, and point out some problems with previous results. We identify several remaining open problems.Comment: A revised version. 16 pages. A preliminary version of this paper appeared as A. C. Cem Say, Abuzer Yakary{\i}lmaz, and \c{S}efika Y\"{u}zsever. Quantum one-way one-counter automata. In R\={u}si\c{n}\v{s} Freivalds, editor, Randomized and quantum computation, pages 25--34, 2010 (Satellite workshop of MFCS and CSL 2010

Describing classical spin Hamiltonians as automata: a new complexity measure

We describe classical spin Hamiltonians as automata and use the classification of the latter to obtain a new complexity measure of Hamiltonians. Specifically, we associate a classical spin Hamiltonian to the formal language consisting of pairs of spin configurations and the corresponding energy, and classify this language in the Chomsky hierarchy. We prove that the language associated to (i) effectively zero-dimensional spin Hamiltonians is regular, (ii) local one-dimensional (1D) spin Hamiltonians is deterministic context-free, (iii) local two-dimensional (2D) or higher-dimensional spin Hamiltonians is context-sensitive, and (iv) totally unbounded spin Hamiltonians is recursively enumerable. It follows that only highly non-physical spin Hamiltonians [(iv)] correspond to Turing machines. It also follows that the Ising model without fields is easy or hard if defined on a 1D or 2D lattice, in contrast to the computational complexity of its ground state energy problem, where the threshold is found between planar and non-planar graphs. Our work puts classical spin Hamiltonians at the same level as automata, and paves the road toward a rigorous comparison of universal spin models and universal Turing machines.Comment: v3: more results; 24 pages, 9 figures, 9 tables. v2: More results and extensively rewritten; 18 pages and 7 figures; code of linear bounded automaton also attached. v1: 13 pages, 7 figures, code of deterministic pushdown automaton attache