Generalized Partitioned Quantum Cellular Automata and Quantumization of Classical CA

J. Watrous introduced the notion of quantum cellular automata(QCA) and showed that any quantum Turing machine can be efficiently simulated by a QCA with constant slowdown in 1995. CA with quantum cells is well-formed QCA if and only if its global transition function is unitary. Generally quantumization of cells of a classical CA dose not make it become QCA, because usually classical CA dose not have reversibility. Morita and Harao show that we can get reversible CA by partition a cell into three part and partitioned CA(PCA) can simulate non-partitioned CA(NPCA)[10]. But there is not a trivial inclusion relation between PCA and NPCA. In this paper, we introduce a new formulation of finite cyclic QCA and generalized notion of partitioned QCA in order to investigate natural transformations from discrete CA to QCA. According to the formulations, we demonstrate a condition of a local transition function, which induce a well-formed QCA. A natural correspondence of classical cells and quantum cells can be extended to the correspondence of classical CA and QCA. If a classical CA satisfies our conditions then the extended QCA is well-formed. Finally we report some results of computer simulations of quantumization of classical CA.