Characterizations of one-way general quantum finite automata

Generally, unitary transformations limit the computational power of quantum finite automata (QFA). In this paper we study a generalized model named one-way general quantum finite automata (1gQFA), in which each symbol in the input alphabet induces a trace-preserving quantum operation, instead of a unitary transformation. Two different kinds of 1gQFA will be studied: measure-once one-way general quantum finite automata (MO-1gQFA) where a measurement deciding to accept or reject is performed at the end of a computation, and measure-many one-way general quantum finite automata (MM-1gQFA) where a similar measurement is performed after each trace-preserving quantum operation on reading each input symbol. We characterize the measure-once model from three aspects: the closure property, the language recognition power, and the equivalence problem. Specially, we prove that MO1gQFA recognize, with bounded error, precisely the set of all regular languages. Our results imply that some quantum finite automata proposed in the literature, which were expected to be more powerful, still can not recognize non-regular languages. We prove that MM-1gQFA also recognize only regular languages with bounded error. Thus, MM-1gQFA and MO-1gQFA have the same language recognition power, which is greatly different from the conventional case in which the number of times the measurement is performed in the computation generally affects the language recognition power of one-way QFA. Finally, we present a sufficient and necessary condition for two MM-1gQFA to be equivalent. ∗This work is supported in part by the National Natural Science Foundation (Nos. 60873055, 61073054), the Natural Science Foundation of Guangdong Province of China (No. 10251027501000004), the Fundamental Research Funds for the Central Universities (No. 10lgzd12), the Program for New Century Excellent Talents in University (NCET) of China, and the project of SQIG at IT, funded by FCT and EU FEDER projects Quantlog POCI/MAT/55796/2004 and QSec PTDC/EIA/67661/2006, IT Project QuantTel, NoE Euro-NF, and the SQIG LAP initiative. Lvzhou Li is partly supported by the China Postdoctoral Science Foundation funded project (20090460808).