Maximum Independent Set via Probabilistic and Quantum Cellular Automata

TL;DR

This paper explores probabilistic and quantum cellular automata as novel frameworks for efficiently solving the Maximum Independent Set problem, demonstrating their potential to outperform traditional quantum optimization methods.

Contribution

It introduces a new PCA and QCA approach for MIS, showing how these automata can converge to solutions and scale with graph size, offering an alternative to existing quantum algorithms.

Findings

MIS convergence probability increases with activation probability p

QCA concentrates population on MIS configurations through cycles

Scaling analysis suggests QCA as an efficient optimization method

Abstract

We study probabilistic cellular automata (PCA) and quantum cellular automata (QCA) as frameworks for solving the Maximum Independent Set (MIS) problem. We first introduce a synchronous PCA whose dynamics drives the system toward the manifold of maximal independent sets. Numerical evidence shows that the MIS convergence probability increases significantly as the activation probability p tends to 1, and we characterize how the steps required to reach the absorbing state scale with system size and graph connectivity. Motivated by this behavior, we construct a QCA combining a pure dissipative phase with a constraint-preserving unitary evolution that redistributes probability within this manifold. Tensor Network simulations reveal that repeated dissipative--unitary cycles concentrate population on MIS configurations. We also provide an empirical estimate of how the convergence time scales with graph size, suggesting that QCA dynamics can provide an efficient alternative to adiabatic and variational quantum optimization methods based exclusively on local and translationally invariant rules.