@MISC{_10quantum, author = {}, title = {10 Quantum Complexity Theory I}, year = {} }

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Abstract

Just as the theory of computability had its foundations in the Church-Turing thesis, computational complexity theory rests upon a modern strengthening of this thesis, which asserts that any “reasonable ” model of computation can be efficiently simulated on a probabilistic Turing machine (by “efficient ” we mean here a runtime that is bounded by a polynomial in the runtime of the simulated machine). For example, computers that can operate on arbitrary length words in unit time, or that can exactly compute real numbers with infinite precision are unreasonable models, since it seems clear that they cannot be physically implemented. It had been argued that the Turing machine model is the inevitable choice once we assume that we can implement only finite precision computational primitives. Given the widespread belief that NP � = BPP, this would seem to put a wide range of important computational problems (the NP-hard problems) well beyond the capability of computers. However, the Turing machine is an inadequate model for all physically realizable computing devices for a fundamental reason: the Turing machine is based on a classical physics model of the universe, whereas current physical theory asserts that the universe is quantum physical. Can we get inherently new kinds of (discrete) computing devices based on quantum physics? The first