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

Abstract

h i g h l i g h t s• We use an information heat engine to physically create a permutation operation.• The mechanism relies on Bennett's algorithm of the reversible Turing machine. • Reversion of the algorithm provoked erasure of information with gain of entropy.• Erasure with gain of entropy causes a one-way (permutation) quantum gate.• We showed that a cascade of two C-NOT gates in an adiabatic constraint is one-way. a b s t r a c tCurrently, 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 noninvertible 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.

Quantum one-way permutation over the finite field of two elements

“…It is straightforward to see that strongly non-invertible functions are invertible from the definition itself of one-way functions. (See a thermodynamic approach of one wayness [32] in input-saving machines [33,34]).…”

Revisiting thermodynamics in computation and information theory

“…The problem that one encounters is the closing of the cycle. An approach to solve this problem has been proposed in [159]. The approach one can think of is by moving the partition wall to the extreme right of the chamber as shown in Fig.…”

“…The existence of one-way-ness has a great impact on the complexity class problems, as one can link this to the physical constraints rather than its mathematical limitations [32,43,44]. In the seminal work [159], they have proposed an approach to bridge a connection between the one-way permutation to thermodynamic computation. They have shown that a quantum circuit maps a target bit to itself [45].…”