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One-way-ness in the input-saving (Turing) machine

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  • de Castro, Alexandre

Abstract

Currently, a complexity-class problem is proving the existence of one-way permutations: one-to-one and onto maps that are computationally ‘easy’, while their inverses are computationally ‘hard’. In what follows, we make use of Bennett’s algorithm of the reversible Turing machine (quantum information heat engine) to perform a cascade of two controlled-NOT gates to physically create a permutation operation. We show that by running this input-saving (Turing) machine backwards the critical inequality of Landauer’s thermodynamic limit is reversed, which provokes the symmetry-breaking of the quantum circuit based on two successive controlled-NOT quantum gates. This finding reveals that a permutation of controlled-NOT gates becomes one-way, provided that adiabatically immersed in a heat bath, which determines the condition of existence of a thermodynamically non-invertible bijection in polynomial-time, that would otherwise be mathematically invertible. This one-way bijection can also be particularly important because it shows nonlinearities in quantum mechanics, which are detectable by watching that the mathematical reversibility of controlled-NOT gates does not work physically.

Suggested Citation

  • de Castro, Alexandre, 2014. "One-way-ness in the input-saving (Turing) machine," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 415(C), pages 473-478.
    • Handle: RePEc:eee:phsmap:v:415:y:2014:i:c:p:473-478
    DOI: 10.1016/j.physa.2014.08.021
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    References listed on IDEAS

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    1. Lídia del Rio & Johan Åberg & Renato Renner & Oscar Dahlsten & Vlatko Vedral, 2011. "The thermodynamic meaning of negative entropy," Nature, Nature, vol. 476(7361), pages 476-476, August.
    2. Landauer, Rolf, 1999. "Information is a physical entity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 263(1), pages 63-67.
    3. Lídia del Rio & Johan Åberg & Renato Renner & Oscar Dahlsten & Vlatko Vedral, 2011. "The thermodynamic meaning of negative entropy," Nature, Nature, vol. 474(7349), pages 61-63, June.
    4. de Castro, Alexandre, 2005. "A network model for clonal differentiation and immune memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 355(2), pages 408-426.
    5. Landauer, Rolf, 1993. "Statistical physics of machinery: forgotten middle-ground," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 194(1), pages 551-562.
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