Don Winter, Thiago L. M. Guedes, Markus Müller
Apr 23, 2026·quant-ph·PDF Execution of quantum algorithms on large-scale quantum computers will require extremely low logical error rates, which necessitates the development of scalable decoding architectures. Local decoders are promising candidates for this task, as they avoid the communication and data processing bottlenecks inherent in global decoding strategies. Cellular automaton (CA) decoders represent a distinct class of local decoders, offering a path toward the low-latency, real-time decoding required for practical applications. In this work, we present SCALA (Signaling CA with Local Attraction), a novel non-hierarchical cellular automaton decoder for quantum repetition and toric codes. By evaluating SCALA alongside the hierarchical CA decoder proposed by Harrington, we provide a direct comparison between non-hierarchical and renormalization-group-style local decoding strategies. We characterize SCALA across three key metrics: Performance, scalability, and robustness. Our results show that SCALA achieves a code-capacity threshold of approximately $p_c\approx 7.5\%$ and provides strong sub-threshold scaling of about $p_L\propto p^{d/4}$ on the toric code. In terms of scalability, our non-hierarchical design ensures that the local computational resources remain independent of system size, yielding a modular local architecture suitable for hardware implementation. Finally, SCALA demonstrates strong robustness to qubit measurement errors and noise within the decoder itself, a critical advantage for real-time decoding on noisy hardware. Our results establish SCALA as a high-performance, scalable, and robust local decoder for scalable quantum error correction.
Terence P Kee, James mcCrum
Previous publications by the authors put forward the argument that Lifelike Cellular Automata can be treated as a bona fide example of livingness in and of themselves, not simply a toy analogue to biological life. Traits known to be indicative of biological life, biosignatures, were identified in informational form as particular outlier traits of the ruleset for the lifelike cellular automata known as Conways Game of Life. This publication reverses that logic, looking at a known outlier trait of Conways Game of Life, its very long-lasting evolutions, and using this to point towards temporal retention as an informational biosignature concept.
Matthew J Simpson, Michael J Plank
Stochastic models of diffusion are routinely used to study dispersal of populations, including populations of animals, plants, seeds and cells. Advances in imaging and field measurement technologies mean that data are often collected across a range of scales, including count data collected across a series of fixed sampling regions to characterize population-level dispersal, as well as individual trajectory data to examine at the motion of individuals within a diffusive population. In this work we consider a lattice-based random walk model and examine the extent to which model parameters can be determined by collecting count data and/or trajectory data. Our analysis combines agent-based stochastic simulations, mean-field partial differential equation approximations, likelihood-based estimation, identifiability analysis, and model-based prediction. These combined tools reveal that working with count data alone can sometimes lead to challenges involving structural non-identifiability that can be alleviated by collecting trajectory data. Furthermore, these tools allow us to explore how different experimental designs impact inferential precision by comparing how different trajectory data collection protocols affects practical identifiability. Open source implementations of all algorithms used in this work are available on GitHub.
Attila Egri-Nagy, Chrystopher L. Nehaniv
Computational power can be measured by assigning an algebraic structure to a computational device. Here, we convert a small patch of Conway's Game of Life into a transformation semigroup. The conversion captures not only time evolution but also interactive operations. In this way, the cellular automaton becomes directly programmable. Once this measurement is made, we apply hierarchical decompositions to the resulting algebraic object as a way of understanding it. These decompositions are based on a macro/micro-state division inspired by statistical mechanics. However, cellular automata have a large number of global states. Therefore, we focus on partitioning the state space and creating morphic images approximations that can serve as macro-level descriptions. The methods developed here are not limited to cellular automata; they apply more generally to discrete dynamical systems.
Vassil Ivanov, Vesselin Tonchev, Marta A. Chabowska, Hristina Popova, Magdalena A. Załuska-Kotur
The coexistence of step bunching and step meandering remains contradictory in the understanding of the unstable step-flow growth. Considered separately, the two instabilities have generated rich but largely independent modeling traditions. Especially, the one-dimensional framework faces a fundamental difficulty once bunching and meandering occur simultaneously -- step bunching is usually associated with an inverted Ehrlich--Schwoebel effect, whereas step meandering is associated with a direct one. The key experiments also focus mainly on the two basic limiting cases. How, then, can both instabilities coexist within the same growth process once the simultaneous occurrence of bunching and meandering cannot be adequately captured as a simple superposition of the two? In this work, we confront results from two substantially different approaches: a (2+1)D Vicinal Cellular Automaton based model (VicCA) and a differential-difference PDE-based description combining a model of step bunching with a relaxation term in the perpendicular direction. The continuous framework enables to explore long-time scales evolution to find large variety of surface patterns. Introducing a proper shape of the potential energy landscape in the VicCA model produces similar patterns and links both models on the level of parameters.
Franco Bagnoli, Luca Mencarelli
We investigate Boolean, totalistic cellular automata with a majority or frustrated majority vote rule, and an interaction range of variable span. These two models show a behavior which differs from the mean-field one. The majority vote model is characterized by the presence of absorbing states, and there is a related bifurcation according to the initial density, in agreement with the mean-field approximation. For initial density equal to $0.5$, however, the dynamics is dominated by a coarsening process, which stops when clusters with a definite curvature radius are established. For the frustrated majority vote model, the mean-field approximation gives chaotic oscillations or a limit cycle. Instead, we observe active patterns, with stable density. Above a certain critical value for the interacting radius there is a bifurcation of the asymptotic density as a function of the initial one.
Franco Bagnoli, Sara Dridi, Bassem Sellami, Amira Mouakher, Samira El Yacoubi
In mathematics and engineering, control theory is concerned with the analysis of dynamical systems through the application of suitable control inputs. One of the prominent problems in control theory is controllability which concerns the ability to determine whether there exists a control input that can steer a dynamical system from an initial state to a desired final state within a finite time horizon. There is a general theory for controlling linear or linearizable system, but it cannot be applied to discrete systems like cellular automata, which is the problem of that we address in this paper. We develop a general theory for linear (and affine) cellular automata, and apply it to examples of one-dimensional and two-dimensional Boolean cases. We introduce the concept of controllability matrix and show that controllability holds if and only if the controllability matrix is invertible.
Franco Bagnoli, Bassem Sellami, Amira Mouakher, Samira El Yacoubi
In this exploratory paper we introduce the problem of cognitive agents that learn how to modify their environment according to local sensing to reach a global goal. We concentrate on discrete dynamics (cellular automata) on a two-dimensional system. We show that agents may learn how to approximate their goal when the environment is passive, while this task becomes impossible if the environment follows an active dynamics.
Fatiha Hamdi, Abdelhafid Zeroual, Fouzi Harrou
Hybrid physical systems combine continuous and discrete dynamics, which can be simultaneously affected by faults. Conventional fault detection methods often treat these dynamics separately, limiting their ability to capture interacting fault patterns. This paper proposes a unified fault detection framework for hybrid dynamical systems by integrating an Extended Timed Continuous Petri Net (ETCPN) model with semi-supervised anomaly detection. The proposed ETCPN extends existing Petri net formalisms by introducing marking-dependent flow functions, enabling intrinsic coupling between discrete and continuous dynamics. Based on this structure, a mode-dependent hybrid observer is designed, whose stability under arbitrary switching is ensured via Linear Matrix Inequalities (LMIs), solved offline to determine observer gains. The observer generates residuals that reflect discrepancies between the estimated and measured outputs. These residuals are processed using semi-supervised methods, including One-Class SVM (OC-SVM), Support Vector Data Description (SVDD), and Elliptic Envelope (EE), trained exclusively on normal data to avoid reliance on labeled faults. The framework is validated through simulations involving discrete faults, continuous faults, and hybrid faults. Results demonstrate high detection accuracy, fast convergence, and robust performance, with OC-SVM and SVDD providing the best trade-off between detection rate and false alarms. The framework is computationally efficient for real-time deployment, as the main complexity is confined to the offline LMI design phase.
Manuel Pita
Cellular automata generate spatially extended, temporally persistent emergent structures from local update rules. No general method derives the mechanisms of that generation from the rule itself; existing tools reconstruct structure from observed dynamics. This paper shows that the look-up table contains a readable causal architecture and introduces a forward model to extract it. The key observation in elementary cellular automata (ECA) is that adjacent cells share input positions, so the prime implicants of neighbouring transitions overlap. That overlap can couple the transitions causally or leave them independent. We formalize each pairwise interaction as a tile. A finite-state, tiling transducer, $\mathcal{T}$, composes tiles across the CA lattice, tracking how coupling and independence propagate from one cell pair to the next. Structural properties of $\mathcal{T}$ are used to classify ECA rules that can sustain regions of causal independence across space and time. We find that, in the 88 ECA equivalence classes, the number of local configurations at which coupling is structurally impossible -- computable from the look-up table -- predicts the prevalence of dynamically decoupled regions with Spearman $ρ= 0.89$ ($p < 10^{-31}$). The look-up table encodes not just what a rule computes but where it distributes causal coupling across the lattice; the framework reads that distribution forward, from local logical redundancy to emergent mesoscopic organization.
E. Chan-López, A. Martín-Ruiz
A comparative algebraic framework for elementary cellular automata is developed, centered on the role of spatial symmetry. The primary object of study is Rule~22, the elementary cellular automaton with algebraic normal form $g(a,b,c)=a\oplus b\oplus c\oplus abc$ over $\mathcal{F}_2$, the simplest rule combining full $S_3$ symmetry with genuine nonlinearity. Three closed-form results are established: a formula for the support-set cardinality, $|S_m|=2^{\mathrm{popcount}(\lfloor m/2 \rfloor)}\cdot 3^{m\bmod 2}$; a two-step recursive construction of the support sets; and the continuous limit as a parabolic reaction--diffusion equation, $\partial_m u=u_{xx}+2u+u^3$. Rule~22 is then used as a symmetric reference for Rule~30. The symmetry-breaking deviation $ε(m)=|S_m^{(30)}|-|S_m^{(22)}|$ is empirically consistent with a power-law scaling of the form $m^b$ ($b\approx 1.11$), quantifying the cumulative effect of replacing the symmetric cubic $abc$ with the asymmetric quadratic $bc$. A mechanism for the apparent randomness of Rule~30's center column is identified through the left-permutive structure and asymmetric Boolean sensitivity profile.
Francisco J. Muñoz, Juan Carlos Nuño
We study one dimensional binary Probabilistic Cellular Automaton (PCA) that interpolate between Wolfram's classical rules 23, 77, 178 and 232. These rules are the only ones that satisfy two criteria: (i) in the case of a majority in the neighborhood states, the central site takes either the majority state or the opposite and (ii) if the neighborhood states are tied, the central site either changes its current state or keeps it. The PCA is defined by two Bernoulli random variables with parameters $p,r \in [0,1]$, and we analytically solve small size cases by using a Markov process formulation. We derive analytical expressions for the probability of asymptotically reaching each possible global configuration as a function of $p$ and $r$, for all initial states. We show that for $0 < p,r < 1$, the asymptotic probability distributions of achieving any of the states for the PCA are independent of the initial conditions. This contrasts with the behavior of the deterministic Wolfram's rules 23 ($p=0,r=0$), 77 ($p=1,r=0$), 178 ($p=0,r=1$) and 232 ($p=1,r=1$), for which additional asymptotic states can occur, in particular periodic configurations.
Adam Nahum, Sthitadhi Roy
Deterministic classical cellular automata can be in two phases, depending on how irreversible the dynamical rules are. In the strongly irreversible phase, trajectories with different initial conditions coalesce quickly, while in the weakly irreversible phase, trajectories with different initial conditions can remain different for a time exponential in the system volume. The transition between these phases is referred to as the damage-spreading transition (the "damaged" sites are those that differ between the trajectories). We develop a theory for this transition. In the simplest and most generic setting, the transition is known to be related to directed percolation, one of the best-studied nonequilibrium phase transitions. However, we show that full theory of the damage-spreading critical point is richer than directed percolation, and contains an infinite hierarchy of sectors of local observables. Directed percolation describes the first level of the hierarchy. The higher observables include "overlaps" for multiple trajectories, and may be labeled by set partitions. (These higher observables arise naturally if, for example, we consider decay of entropy under the irreversible dynamics.) The full hierarchy yields a hierarchy of nonequilibrium fixed points for reaction-diffusion-type processes, all of which contain directed percolation as a subsector, but which possess additional universal critical exponents. We analyze these higher fixed points using a field theory formulation and renormalization group arguments, and using simulations in 1+1 dimensions.
Chaoqian Wang, Jingyang Li, Xinwei Wang, Wenqiang Zhu, Attila Szolnoki
Cooperating first then mimicking the partner's act has been proven to be effective in utilizing reciprocity in social dilemmas. However, the extent to which this, called Tit-for-Tat strategy, should be regarded as equivalent to unconditional cooperators remains controversial. Here, we introduce a biased Tit-for-Tat (T) strategy that cooperates differently toward unconditional cooperators (C) and fellow T players through independent bias parameters. The results show that, even under strong dilemmas in the donation game framework, this three-strategy system can exhibit diverse phase diagrams on the parameter plane. In particular, when T-bias is small and C-bias is large, a ``hidden T phase'' emerges, in which the weakest T strategy dominates. The dominance of the weakened T strategy originates from a counterintuitive mechanism characterizing non-transitive ecological systems: T suppresses its relative fitness to C, rapidly eliminates the cyclic dominance clusters, and subsequently expands slowly to take over the entire population. Analysis in well-mixed populations confirms that this phenomenon arises from structured populations. Our study thus reveals the subtle role of bias regulation in cooperative modes by emphasizing the ``survival of the weakest'' effect in a broader context.
Xuan Kien Phung
Gottschalk's surjunctivity conjecture states that for all group universes and finite alphabets, every equivariant and continuous selfmap of the full shift, known as cellular automaton, cannot be a strict embedding. Not all surjective cellular automata are injective. However, if the surjectivity condition is replaced by a certain strengthened property called post-surjectivity then all post-surjective cellular automata must be bijective whenever the universe is a sofic group. A group universe is said to be post-injunctive if every post-surjective cellular automaton with finite alphabet over this group universe must be bijective. Gromov's injectivity lemma states each injective cellular automaton over a subshift can be extended to an injective cellular automaton over every subshift which is close enough to the initial subshift. In this paper, we obtain analogous results where injectivity is replaced by other fundamental dynamical properties namely post-surjectivity and pre-injectivity. We also study various stable properties of the class of post-injunctive groups in parallel to properties of surjunctive groups. Among the results, we show that semidirect extensions of post-injunctive groups with residually finite kernels must be post-injunctive.
Haidong Zhang, Chaoqian Wang, Shuo Liu, Charo I. del Genio, Stefano Boccaletti, Xin Lu
Trust is one of the cornerstones of human society. One of the evolutionary pressure mechanisms that may have led to its emergence is the presence of incentives for trustworthy behavior. However, this type of reward has received relatively little attention in the context of spatial trust games, which are often used to build models in evolutionary game theory. To fill this gap, we introduce an inter-role reward mechanism in the spatial trust game, so that an investing trustor can choose to pay an extra cost to reward a trustworthy trustee. With extensive numerical simulations, we find that this type of reward does not always promote trust. Rather, while moderate rewards break the dominance of mistrust, thereby favoring investment, excessive rewards eventually stimulate a nonreturn strategy, ultimately suppressing the evolution of trust. Additionally, lower reward costs do not necessarily promote trust. Instead, more costly, but not excessive, rewards enhance the advantage of the original investment, consolidating the clusters of rewarders and improving trust. Our model thus provides evidence about the counterintuitive nature of the relationship between trust and rewards in a complex society.
Louis Paletta
Local decoders provide a promising approach to real-time quantum error-correction by replacing centralized classical decoding, with significant hardware constraints, by a fully distributed architecture based on a simple, local update rule. We propose a new local decoder for Kitaev's toric code: the 2D signal-rule, that interprets odd parity stabilizer measurements as defects, attracted to each other via the exchange of binary signals. We present numerical evidence of exponential suppression of the logical error rate with system size below a threshold, under a phenomenological noise model with data and measurement errors at each iteration. The construction achieves a significantly improved threshold and optimal finite-size scaling relative to hierarchical schemes. It also provides a lightweight alternative to windowed local decoder constructions while maintaining strong performance, thus enabling a streamlined architecture for a two-dimensional local quantum memory.
Thomas M. A. Fink
Input-output maps are prevalent throughout science and technology. They are empirically observed to be biased towards simple outputs, but we don't understand why. To address this puzzle, we study the archetypal input-output map: a deep-layered machine in which every node is a Boolean function of all the nodes below it. We give an exact theory for the distribution of outputs, and we confirm our predictions through extensive computer experiments. As the network depth increases, the distribution becomes exponentially biased towards simple outputs. This suggests that deep-layered machines and other learning methodologies may be inherently biased towards simplicity in the models that they generate.
Aoi Araoka, Tetsuji Tokihiro
This paper explores cellular automata (CA) constructed from Yang-Baxter maps over finite fields $F_{2^n}$. We define $R$-matrices using a map $f$ on $F_{2^n}$ and establish necessary and sufficient conditions for $f$ to satisfy the Yang-Baxter equation. We show that these conditions become remarkably streamlined in characteristic two. An exhaustive search for bijective solutions in fields of order 4, 8, and 16 yields 16, 736, and 269,056 maps, respectively. Analysis of the resulting CA under helical boundary conditions reveals a consistent alignment between the temporal period and the field order. We propose the conjecture that this periodic identity holds generally for $F_{2^n}$, supported by analytical proofs for $n=2$ and $n=3$. Our results further indicate that bijectivity is a fundamental requirement for this periodic behavior.
Emilio N. M. Cirillo, Joram L. Vliem, Dirk Schuricht, Cristian Spitoni
We study a probabilistic cellular automaton obtained as a mixture of the additive elementary rules 60 and 102. We prove that, for any finite periodic lattice and for mixing parameter $λ=1/2$, the system almost surely reaches the absorbing all-zero configuration in finitely many steps. In addition, Monte Carlo simulations indicate as well the presence of a zero-density stationary state in a finite interval around $λ=1/2$. Despite this absorbing behavior, both mean-field and block approximation schemes predict a stationary state with non-zero density. This failure, traced to the additive and mirror symmetries of the deterministic components, highlights a fundamental limitation of finite-block approximation in capturing the global dynamics of probabilistic cellular automata.