Abstract
The relation between the physical world where the hypothetical quantum computer is supposed to reside, and the formal mathematical theorem proofs is analysed. In mathematics, one proves theorems on the basis of axioms, which by definition are supposed to be fullfilled exactly. However, in the physical world nothing can be exact. The existing theory does not respond to the obvious question: what precision is needed?
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Invented by the English mathematician and clergyman William Oughtred in about 1662 (see: https://en.wikipedia.org/wiki/William_Oughtred), during 3 centuries the slide rule has reliably served to generations of scientists and engineers, before being replaced, only about 50 years ago, by electronic calculators and later—by modern computers.
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In the popular press, the term “exponential” has acquired the meaning of “extraordinary”, “enormous”, etc. In fact, it serves simply to define a function y(x), such that y ~ ax. The difficulty of building a quantum computer increases proportional to the number of continuous parameters that should be under control, which is 2 N, where N is the number of qubits. Indeed, this is an exponential dependence on N. In contrast, a classical analog computer based on N pointers, needs only 2N parameters to be under control.
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I thank Konstantin Dyakonov for providing the text below, and Ekaterina Diakonova for drawing the picture.
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Spin relaxation, i.e. the disappearance of the initial non-equilibrium spin polarization due to various natural causes was extensively studied for a very long time in gases, liquids, and solids. Depending on the system, the temperature, and the experimental conditions, the spin relaxation time τs for electrons usually varies from nanoseconds to milliseconds. As a rule, the decay of the average spin polarization is studied. However, it can be shown that the general nonequilibrium state of a system of N spins decays N times faster. Thus, if the average spin relaxation time is 1 s (which is enormously long!), but your system consists of N = 1000 spins, the initial spin state of the whole system will become unrecognizable after 1 ms. This gives us the upper time limit for performing our entire quantum algorithm.
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Dyakonov, M.I. (2020). The Theoretical Quantum Computer and the Physical Reality. In: Will We Ever Have a Quantum Computer?. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-42019-2_4
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DOI: https://doi.org/10.1007/978-3-030-42019-2_4
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-030-42019-2
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