Time-Space Complexity Advantages for Quantum Computing

It has been proved that quantum computing has advantages in query complexity, communication complexity and also other computing models. However, it is hard to prove strictly that quantum computing has advantage in the Turing machine models in time complexity. For example, we do not know how to prove that Shor's algorithm is strictly better than any classical algorithm, since we do not know the lower bound of time complexity of the factoring problem in Turing machine. In this paper, we consider the time-space complexity and prove strictly that quantum computing has advantages compared to their classical counterparts. We prove: (1) a time-space upper bound for recognition of the languages LINT (n) on two-way finite automata with quantum and classical states (2QCFA): TS = O(n(3/2) log n), whereas a lower bound on probabilistic Turing machine is TS = Omega(n(2)); (2) a time-space upper bound for recognition of the languages L-NE(n) on exact 2QCFA: TS = O(n(1.87) log n), whereas a lower bound on probabilistic Turing machine is TS = Omega(n(2)). It has been proved (Klauck, STOC'00) that the exact one-way quantum finite automata have no advantage comparing to classical finite automata in recognizing languages. However, the result (2) shows that the exact 2QCFA do have an advantage in comparison with their classical counterparts, which is the first example showing that the exact quantum computing has advantage in time-space complexity comparing to classical computing.