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After reading Church's thesis, and godelization theory, it seems everything is futile, is this really true, all the machines we are making machine which is same as before, bounded to not solve halting problems, in more simple terms can quantum computer do anything that a Turing machine can't?

PS: In book named Elements of theory of computation first edition, you will find Godelization otherwise it is redacted since second edition reason unknown but chapter 5 named Church Thesis is highly altered since second edition.

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No. A classical computer can compute anything a quantum computer can do, given long enough. There are simple methods of simulation that cause an exponential slowdown, which suffices to show that if it is decidable on a quantum computer, it is decidable on a classical computer. (We don't know whether the there might be other faster ways of simulating a quantum computer on a classical computer, but that doesn't affect decidability.)

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  • $\begingroup$ So all the hope for GOD machine is futile, if quantum is same as classical, what is the benefit in living a false hope for technology innovation? $\endgroup$ – TheExorcist Nov 30 '20 at 2:55
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    $\begingroup$ @TheExorcist, You asked a technical question. I gave an answer. The question and answer are probably primarily of theoretical interest. They have very little implications about hope for technology innovation. Quantum computers are not a god machine; they never were. $\endgroup$ – D.W. Nov 30 '20 at 5:33
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As far as we know right now, no. Quantum computation makes some problems tractable that are currently intractable for classical computers, but problems that are theoretically impossible for classical Turing machines remain impossible for quantum computers as we understand them. Intractable, meaning the computation would take too long for the result to be useful. Impossible meaning literally not possible.

Computing something outside what's possible for a Turing machine enters the realm of hypercomputation, but that's not quite what quantum computers do.

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