Abstract
special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications We introduce a new model of two-way finite automaton, which is endowed with the capability of resetting the position of the tape head to the left end of the tape in a single move during the computation. Several variants of this model are examined, with the following results: The weakest known model of computation where quantum computers recognize more languages with bounded error than their classical counterparts is identified. We prove that two-way probabilistic and quantum finite automata (2PFAs and 2QFAs) can be considerably more concise than both their one-way versions (1PFAs and 1QFAs), and two-way nondeterministic finite automata (2NFAs). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than 1 2 by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFAs, 1QFAs and 2NFAs grow without bound. We also show that 2QFAs with mixed states can support highly efficient probability amplification.
Highlights
In recent years, the research effort on quantum versions of finite automata has mainly focused on oneway models, with the study of two-way quantum finite automata (2QFAs), which are synonymous with constant space quantum Turing machines, receiving relatively less attention
We show that 2-way quantum finite automaton with reset (2QFA) with mixed states can support highly efficient probability amplification, surpassing the best known methods for 2KWQFAs recognizing these languages
Watrous [Wat97] notes that a 2KWQFA algorithm he presents for recognizing a nonregular language is remarkably costly in terms of probability amplification, and states that this problem stems from the fact that 2KWQFAs cannot “reset” themselves during execution to repeatedly carry out the same computation
Summary
The research effort on quantum versions of finite automata has mainly focused on oneway models, with the study of two-way quantum finite automata (2QFAs), which are synonymous with constant space quantum Turing machines, receiving relatively less attention. The Kondacs-Watrous model of quantum finite automaton (to be called, on, KWQFA), which allows measurements of a restricted type, rather than the full set sanctioned by quantum theory, has been proven to be weaker in terms of language recognition power [KW97], probability amplification capability [AF98], and, in some cases at least, succinctness [ANTSV02], than the corresponding classical model, in the one-way case. More general models, such as the 2QCFA, employing mixed states, are able to simulate the corresponding classical probabilistic automata efficiently in both the one-way and two-way settings, and to recognize some languages that 2PFAs cannot [AW02].
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