Computational power of quantum and probabilistic automata

Elektroniskā versija nesatur pielikumusAnot¹acija ņSis darbs apvieno p¹et¹³jumus par diviem autom¹atu veidiem: varb¹utiskajiem apgrieņzamajiem autom¹atiem (PRA), kas ir saist¹³ti ar kvantu gal¹³gajiem auto- m¹atiem (QFA), un vienvirziena kvantu autom¹atiem ar skait¹³t¹aju (Q1CA), kas ir l»oti ierobeņzots kvantu autom¹atu modelis, kam atbilstoņsa kvantu sist¹ema nav gal¹³ga. Darba m¹erk»is ir aprakst¹³t valodu klases, ko paz¹³st ņsie autom¹ati, un sal¹³dzin¹at kvantu un varb¹utiskos autom¹atus. M¹es pied¹av¹ajam varb¹utisk¹a apgrieņzama autom¹ata modeli. M¹es p¹et¹am vienvirziena PRA gan ar klasisko (C-PRA) v¹ardu akcept¹eņsanu, gan ar ap- st¹adin¹aņsanu (DH-PRA). M¹es par¹ad¹am valodu klases a¤1 a¤2 : : : a¤n paz¹³ņsanu ar PRA. M¹es par¹ad¹am vai valodu klase, ko paz¹³st PRA, ir sl¹egta pret B¹ula oper¹acij¹am. M¹es par¹ad¹am visp¹ar¹³gas valodu klases, ko C-PRA un DH-PRA nepaz¹³st. M¹es apskat¹am v¹ajas apgrieņzam¹³bas de¯n¹³ciju un par¹ad¹am atņsk»ir¹³bu no apgrieņzam¹³bas. M¹es pied¹av¹ajam vispar¹³gu kvantu vienvirziena autom¹ata modeli ar skai- t¹³t¹aju (Q1CA). M¹es pier¹ad¹am ka ņsis modelis apmierina transform¹acijas uni- tarit¹ates principu. Tiek pied¹av¹ats speci¹als Q1CA veids - vienk¹arņsais Q1CA, kas l»auj konstru¹et autom¹atu piem¹erus konkr¹et¹am valod¹am. M¹es par¹ad¹am vair¹aku kontekstatkar¹³go valodu paz¹³ņsanu ar Q1CA. M¹es pier¹ad¹am ka past¹av valodas, ko paz¹³st Q1CA, bet ko nepaz¹³st varb¹utiskais autom¹ats ar skait¹³t¹aju.The thesis assembles research on two models of automata - probabilistic reversible (PRA) that appear very similar to 1-way quantum ¯nite automata (1-QFA) and quantum one-way one counter automata (Q1CA), that is the most restricted model of non-¯nite space quantum automata. The objective of the research is to describe classes of languages recognizable by these models and compare related quantum and probabilistic automata. We propose the model of probabilistic reversible automata. We study both one-way PRA with classical (1-C-PRA) and decide and halt (1-DH- PRA) acceptance. We show recognition of general class of languages Ln = a¤1a¤2 : : : a¤n with probability 1 ¡ ". We show whether the classes of languages they recognize are closed under boolean operations and describe general class of languages not recognizable by these automata in terms of \forbidden con- structions" for the minimal deterministic automaton of the language. We also consider \weak" reversibility as equivalent de¯nition for 1-way automata and show the di®erence from ordinary reversibility in 1.5-way case. We propose the general notion of quantum one-way one counter au- tomata(Q1CA). We describe well-formedness conditions for the Q1CA that ensure unitarity of its evolution. A special kind of Q1CA, called simple, that satis¯es the well-formedness conditions is introduced. We show recognition of several non context free languages by Q1CA. We show that there is a lan- guage that can be recognized by quantum one-way one counter automaton, but not by the probabilistic one counter automaton

Applying causality principles to the axiomatization of probabilistic cellular automata

Cellular automata (CA) consist of an array of identical cells, each of which may take one of a finite number of possible states. The entire array evolves in discrete time steps by iterating a global evolution G. Further, this global evolution G is required to be shift-invariant (it acts the same everywhere) and causal (information cannot be transmitted faster than some fixed number of cells per time step). At least in the classical, reversible and quantum cases, these two top-down axiomatic conditions are sufficient to entail more bottom-up, operational descriptions of G. We investigate whether the same is true in the probabilistic case. Keywords: Characterization, noise, Markov process, stochastic Einstein locality, screening-off, common cause principle, non-signalling, Multi-party non-local box.Comment: 13 pages, 6 figures, LaTeX, v2: refs adde

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

Finite automata with advice tapes

We define a model of advised computation by finite automata where the advice is provided on a separate tape. We consider several variants of the model where the advice is deterministic or randomized, the input tape head is allowed real-time, one-way, or two-way access, and the automaton is classical or quantum. We prove several separation results among these variants, demonstrate an infinite hierarchy of language classes recognized by automata with increasing advice lengths, and establish the relationships between this and the previously studied ways of providing advice to finite automata.Comment: Corrected typo

Quantum Branching Programs and Space-Bounded Nonuniform Quantum Complexity

In this paper, the space complexity of nonuniform quantum computations is investigated. The model chosen for this are quantum branching programs, which provide a graphic description of sequential quantum algorithms. In the first part of the paper, simulations between quantum branching programs and nonuniform quantum Turing machines are presented which allow to transfer lower and upper bound results between the two models. In the second part of the paper, different variants of quantum OBDDs are compared with their deterministic and randomized counterparts. In the third part, quantum branching programs are considered where the performed unitary operation may depend on the result of a previous measurement. For this model a simulation of randomized OBDDs and exponential lower bounds are presented.Comment: 45 pages, 3 Postscript figures. Proofs rearranged, typos correcte