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We start with a discussion on computational problems and algorithms. We define a generic form of decision problems, and then introduce deterministic finite automaton (DFA) as a basic decider. After having a few examples on the DFAs with discussions on their limitations, we introduce their probabilistic and quantum variants. We present a representative algorithm for each of these variants, followed by a discussion on the quantum advantages in finite automata settings. We shortly review the recent implementations of QFA algorithms on real quantum hardware.