Quantum alternation

We introduce the concept of quantum alternation as a generalization of quantum nondeterminism. We define the first quantum alternating Turing machine (qATM) by augmenting alternating Turing machine (ATM) with a fixed-size quantum register. We focus on space-bounded computation, and obtain the following surprising result: One-way qATMs with constant-space (one-way alternating quantum finite automata (1AQFAs)) are Turing-equivalent. Then, we introduce strong version of qATM: The qATM that must halt in every computation path. We show that strong qATMs (similar to private ATMs) can simulate deterministic space with 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 interesting results: (i) any language recognized by a rtAQFA is in quadratic deterministic space, (ii) two-alternation is better than one-alternation, (iii) two-alternation is sufficient to recognize a NP-complete language and so any language in NP can be recognized by a poly-time log-space qATM with two alternations, (iv) three-alternation is sufficient to recognize a language that is complete for the second level of the polynomial hierarchy and so any language in the second level of the polynomial hierarchy can be recognized by a poly-time log-space qATM with three alternations.