Reversible cellular automata in presence of noise rapidly forget everything

TL;DR

This paper demonstrates that reversible cellular automata subjected to noise rapidly lose all information about their initial state, emphasizing the necessity of irreversibility for reliable computation under noisy conditions.

Contribution

It proves that reversible cellular automata with additive noise forget initial information exponentially fast, highlighting the importance of irreversibility for scalable noisy computation.

Findings

Finite cell states become indistinguishable from noise after O(log n) steps

Reversible automata cannot reliably preserve information in noisy environments

Irreversibility appears necessary for scalable noise-resilient computation

Abstract

We consider reversible and surjective cellular automata perturbed with noise. We show that, in the presence of positive additive noise, the cellular automaton forgets all the information regarding its initial configuration exponentially fast. In particular, the state of a finite collection of cells with diameter n becomes indistinguishable from pure noise after O(log n) time steps. This highlights the seemingly unavoidable need for irreversibility in order to perform scalable reliable computation in the presence of noise.