Maximum Independent Set via Probabilistic and Quantum Cellular Automata

TL;DR

We address MIS on graphs using probabilistic and quantum cellular automata (PCA and QCA) to harness locality and parallelism for combinatorial optimization. The approach combines a synchronous PCA whose absorbing states are maximal independent sets with a quantum protocol that alternates dissipative relaxation and constraint-preserving unitary updates, biasing probability toward larger MIS configurations. Classical simulations reveal how the MIS convergence probability and convergence time scale with graph size and connectivity, while tensor-network-based quantum simulations quantify the dissipative-to-unitary dynamics and reveal a power-law scaling of relaxation times and a polynomial scaling in cycle counts for open chains. The results demonstrate that local, translationally invariant QCA dynamics can offer an efficient, hardware-compatible heuristic for MIS, with potential advantages over adiabatic and variational quantum schemes on unit-disk or Rydberg-like graphs.

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.

Figures (10)

• Figure 1: Illustration of the probabilistic local update rule: if at least one neighbour is active, the cell becomes inactive with probability $1$ (left). If all neighbours are inactive and the cell is inactive, it may remains inactive (center) or it activates (right) with probability $1-p$ and $p$ respectively.
• Figure 2: MIS convergence probability for different values of ($N$,$k$) and for the following values of $p$: 0.8 (a), 0.9 (b), 0.95 (c) and 0.99 (d). Each value is computed as the average on a set of 10 graph instances with same number of nodes $N$ and average degree $k$
• Figure 3: (a) MIS convergence probability $P_{\text{MIS}}$ (a) and convergence time (b) as a function of $p$ for different values of ($N$,$k$). For each ($N$,$k$) pair, we averaged $P_{\text{MIS}}$ and $n$ from a set of 10 graph instances, computed for different values of $p$
• Figure 4: (a) $n$ vs $N$. (b) $n$ vs $k$. Each point is obtained from the average over 100 random graph instances with $p=0.9$. In subfigure (a), the average degree is fixed to $k=3.0$, whereas in subfigure (b) the system size is fixed to $N=50$. Panel (a) is shown in log–log scale, and the scaling of $n$ with $N$ is best captured by a power–law fit. In contrast, in panel (b) the best fit is exponential, which becomes a straight line when plotting $\log(n)$ on the vertical axis (see main text for fit parameters).
• Figure 5: (a) Heatmap of the mean convergence time $\langle T \rangle$ for system sizes $N \in [10,30]$ and average degrees $k \in [1.5,5.0]$. (b) Dependence of $\langle T \rangle$ on $N$ at fixed $k$ (colored lines) with $\pm 1\sigma$ confidence bands. (c) Dependence of $\langle T \rangle$ on $k$ at fixed $N$.