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2016•Lobachevskii Journal of MathematicsRequires access

Quantum alternation

Abuzer Yakaryılmaz

Open publisher page 4 citations

Abstract

We introduce quantum alternation as a generalization of quantum nondeterminism. We define q-alternating Turing machine (qATM) by augmenting alternating Turing machine with constant-size quantum memory. We show that one-way constant-space qATMs (1AQFAs) are Turing equivalent. Then, we introduce strong version of qATM by requiring to halt in every computation path and we show that strong qATMs can simulate deterministic spacewith exponentially less space. This leads to shifting the deterministic space hierarchy exactly by one level. We also focus on realtime versions of 1AQFAs (rtAQFAs) and obtain many results: rtAQFAs can recognize a PSPACE-complete problem; they cannot be simulated by sublinear deterministic Turing machines; for any level of polynomial hierarchy, say k , there exists a complete language that can be recognized by rtAFAs with only ( k +1) alternations; and polynomial hierarchy lies in its log-space q-alternation counterpart.

About this research paper

What this paper is about

We introduce quantum alternation as a generalization of quantum nondeterminism. We define q-alternating Turing machine (qATM) by augmenting alternating Turing machine with constant-size quantum memory. We show that one-way constant-space qATMs (1AQFAs) are Turing equivalent. Then, we introduce strong version of qATM by requiring to halt in every computation path and we show that strong qATMs can simulate deterministic spacewith exponentially less space. This leads to shifting the deterministic space hierarchy exactly by one level. We also focus on realtime versions of 1AQFAs (rtAQFAs) and obtain many results: rtAQFAs can recognize a PSPACE-complete problem; they cannot be simulated by sublinear deterministic Turing machines; for any level of polynomial hierarchy, say k , there exists a complete language that can be recognized by rtAFAs with only ( k +1) alternations; and polynomial hierarchy lies in its log-space q-alternation counterpart.

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Available abstract

We introduce quantum alternation as a generalization of quantum nondeterminism. We define q-alternating Turing machine (qATM) by augmenting alternating Turing machine with constant-size quantum memory. We show that one-way constant-space qATMs (1AQFAs) are Turing equivalent. Then, we introduce strong version of qATM by requiring to halt in every computation path and we show that strong qATMs can simulate deterministic spacewith exponentially less space. This leads to shifting the deterministic space hierarchy exactly by one level. We also focus on realtime versions of 1AQFAs (rtAQFAs) and obtain many results: rtAQFAs can recognize a PSPACE-complete problem; they cannot be simulated by sublinear deterministic Turing machines; for any level of polynomial hierarchy, say k , there exists a complete language that can be recognized by rtAFAs with only ( k +1) alternations; and polynomial hierarchy lies in its log-space q-alternation counterpart.

Key concepts: Time hierarchy theorem, PSPACE, Alternation (linguistics), Mathematics, Turing machine, Polynomial hierarchy, Probabilistic Turing machine, NSPACE

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