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SOME LANGUAGES RECOGNIZED BY TWOWAY FINITE AUTOMATA WITH QUANTUM AND CLASSICAL States
, 2011
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Exponentially more concise quantum recognition of nonRMM regular languages
"... We show that there are quantum devices that accept all regular languages and that are exponentially more concise than deterministic finite automata (DFA). For this purpose, we introduce a new computing model of oneway quantum finite automata (1QFA), namely, oneway quantum finite automata together ..."
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Cited by 4 (3 self)
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We show that there are quantum devices that accept all regular languages and that are exponentially more concise than deterministic finite automata (DFA). For this purpose, we introduce a new computing model of oneway quantum finite automata (1QFA), namely, oneway quantum finite automata together with classical states (1QFAC), which extends naturally both measureonce 1QFA and DFA and whose state complexity is upperbounded by both. The original contributions of the paper are the following. First, we show that the set of languages accepted by 1QFAC with bounded error consists precisely of all regular languages. Second, we prove that 1QFAC are at most exponentially more concise than DFA. Third, we show that the previous bound is tight for families of regular languages that are not recognized by measureonce (RMO), measuremany (RMM) and multiletter 1QFA. Fourth, we give a polynomialtime algorithm for determining whether any two 1QFAC are equivalent. Finally, we show that the state minimization of 1QFAC is decidable within EXPSPACE. We conclude the paper by posing some open problems.
TimeSpace Efficient Simulations of Quantum Computations
, 2010
"... We give two time and spaceefficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that ev ..."
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We give two time and spaceefficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that every language solvable by a boundederror quantum algorithm running in time t and space s is also solvable by an unboundederror randomized algorithm running in time O(t · log t) and space O(s + log t), as well as by a boundederror quantum algorithm restricted to use an arbitrary universal set and running in time O(t · polylog t) and space O(s + log t), provided the universal set is closed under adjoint. We also develop a quantum model that is particularly suitable for the study of general computations with simultaneous time and space bounds. As an application of our randomized simulation, we obtain the first nontrivial lower bound for general quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are the problems of determining the truth value of a given Boolean formula whose variables are fully quantified by one or two majority quantifiers, respectively. We prove that for every real d and every positive real δ there exists a real c> 1 such that either • MajMajSAT does not have a boundederror quantum algorithm running in time O(n c), or • MajSAT does not have a boundederror quantum algorithm running in time O(n d) and space O(n 1−δ). In particular, MajMajSAT does not have a boundederror quantum algorithm running in time O(n 1+o(1) ) and space O(n 1−δ) for any δ> 0. Our lower bounds hold for any reasonable uniform model of quantum computation, in particular for the model we develop. 1
2012 Dieter van Melkebeek and Thomas Watson Licensed under a Creative Commons Attribution License
, 2011
"... Abstract: We give two time and spaceefficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations sh ..."
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Abstract: We give two time and spaceefficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that every language solvable by a boundederror quantum algorithm running in time t and space s is also solvable by an unboundederror randomized algorithm running in time O(t · logt) and space O(s + logt), as well as by a boundederror quantum algorithm restricted to use an arbitrary universal set and running in time O(t · polylogt) and space O(s + logt), provided the universal set is closed under adjoint. We also develop a quantum model that is particularly suitable for the study of general computations with simultaneous time and space bounds. As an application of our randomized simulation, we obtain the first nontrivial lower bound for general quantum algorithms solving problems related to satisfiability. Our bound applies to MAJSAT and MAJMAJSAT, which are the problems of determining the truth