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arxiv: 2605.13379 · v1 · pith:RAI6DYA2new · submitted 2026-05-13 · 🪐 quant-ph · cond-mat.str-el· cs.FL· nlin.CG

Universal Design and Physical Applications of Non-Uniform Cellular Automata on Translationally Invariant Lattices

Xiang-You Huang , Jie-Yu Zhang , Peng Ye This is my paper

Pith reviewed 2026-05-14 18:10 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-elcs.FLnlin.CG
keywords non-uniform cellular automatahyperbolic latticesubsystem symmetrytopological statesdirected percolationquantum cellular automatatranslation invariance
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The pith

A non-uniform cellular automata algorithm generates subsystem symmetry-protected topological states on hyperbolic lattices unattainable on Euclidean ones.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a higher-order non-uniform cellular automata method that adapts to translationally invariant lattices with constant negative curvature. By embedding geometric information directly into the update rules, it overcomes the incompatibility of standard cellular automata with hyperbolic geometry. This enables creation of new phases of matter, such as subsystem symmetry-protected topological states with irregular symmetries, and simulation of percolation processes that follow the lattice's tree-like structure. Readers interested in how geometry influences quantum matter would find this relevant as it opens hyperbolic space as a platform for exotic physical phenomena.

Core claim

We develop a higher-order non-uniform cellular automata (NUCA) algorithm applicable to both translationally invariant regular Euclidean and hyperbolic lattices. In the algorithm, the non-uniform update rules incorporate nontrivial geometric data through a lattice-deforming procedure. By applying a linear NUCA, we generate subsystem symmetry-protected topological (SSPT) states and spontaneous subsystem symmetry-breaking states associated with regular or irregular subsystem symmetries unattainable on Euclidean lattices.

What carries the argument

The lattice-deforming procedure that incorporates geometric data of the hyperbolic lattice into non-uniform update rules of the cellular automata, preserving translation invariance.

If this is right

  • Application of linear NUCA produces SSPT states on the hyperbolic {5,4} lattice.
  • Multi-point strange correlators are designed to detect these nontrivial SSPT states.
  • A sufficient condition is derived for non-Abelian translationally invariant NUCA-generated models.
  • Generalization to non-uniform Clifford quantum cellular automata generates subsystem symmetries in the hyperbolic cluster state.
  • Probabilistic NUCA simulates directed percolation on the lattice, allowing estimation of percolation thresholds and phase diagrams.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The NUCA framework could be tested on other negatively curved lattices like {7,3} to see if similar symmetry phenomena emerge.
  • This approach might inform designs for quantum simulators on hyperbolic lattices in experimental setups using circuit lattices or optical systems.
  • Extending the method to time-dependent or inhomogeneous geometries could reveal dynamic phase transitions unique to curved spaces.
  • The correspondence between CQCA and subsystem symmetries suggests broader classifications of topological phases in non-Euclidean settings.

Load-bearing premise

The lattice-deforming procedure correctly folds nontrivial geometric data into the non-uniform update rules while preserving the required translation invariance and without introducing spurious correlations or breaking the intended symmetry protection.

What would settle it

If the generated states on the hyperbolic lattice do not satisfy the predicted subsystem symmetries when measured with the multi-point strange correlators, or if the percolation thresholds deviate significantly from expected values due to the treelike structure, the claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.13379 by Jie-Yu Zhang, Peng Ye, Xiang-You Huang.

Figure 1
Figure 1. Figure 1: Schematic overview of the NUCA algorithm. By employing a lattice-deforming procedure that embeds hyperbolic [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of the Euclidean {4, 4} (square) lattice, hyperbolic {5, 4} and {6, 6} lattices, which are studied in de￾tail in this paper. Physical qubits are located on polygons. tem [59, 85–88], we develop a lattice-deforming procedure to embed the hyperbolic lattice into the Euclidean square lattice as shown in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of a semi-infinite Euclidean square [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Visualization of four classes of subsystem symmetries for translationally invariant SSPT models on the Euclidean [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) The splitting method applied to a quarter of the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Visualization of the deformed lattice on which we [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The Hamiltonian and symmetry patterns of the cluster model on the [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Visualization of nontrivial initial condition for the [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The Hamiltonian and symmetry patterns of an SSPT model on the [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Numerical growth of NUCA-generated subsystem symmetries. (a,d) Regular symmetry pattern generated by the [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The Hamiltonian and symmetry patterns generated by the update rule Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The Hamiltonian and symmetry patterns generated by the update rule Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The Hamiltonian and symmetry patterns generated by the update rule Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Derivation of CQCA structure of the cluster state on the [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: A subsystem symmetry of the cluster state on the [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Simulation of directed percolation on the [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Numerical results of directed percolation on the [PITH_FULL_IMAGE:figures/full_fig_p025_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: (a) The splitting method applied to a quarter of the [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: (a) The splitting method applied to a quarter of the [PITH_FULL_IMAGE:figures/full_fig_p035_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Visualization of the deformed lattice where we [PITH_FULL_IMAGE:figures/full_fig_p036_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The Hamiltonian and symmetry patterns of the cluster model on the [PITH_FULL_IMAGE:figures/full_fig_p038_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The bipartite {4, 5} lattice where qubits are located on vertices. The red and blue vertices denote the two sublat￾tices. The model is defined on vertices of the {4, 5} lattice, dual to the polygon model on {5, 4} in Sec. III. The five red (blue) vertices in the red (blue) plaquette with the central blue (red) vertex form the support of a Hamiltonian term. Then we specify the gauge-symmetry terms as: A (a… view at source ↗
read the original abstract

Lattice geometry profoundly shapes physical phenomena such as subsystem symmetry and directed percolation (DP). Among various lattice geometries, hyperbolic lattices are characterized by constant negative curvature and non-Abelian translation symmetry, offering a rich platform for investigating this geometry-physics interplay. However, the exponentially growing lattice size and nontrivial translation symmetry make approaches developed for Euclidean lattices incompatible, a limitation particularly evident in uniform cellular automata (CA). To resolve this, we develop a higher-order non-uniform cellular automata (NUCA) algorithm applicable to both translationally invariant regular Euclidean and hyperbolic lattices. In the algorithm, the non-uniform update rules incorporate nontrivial geometric data through a lattice-deforming procedure. We demonstrate the broad applicability of our algorithm to hyperbolic lattices through several applications on the hyperbolic $\{5,4\}$ lattice. By applying a linear NUCA, we generate subsystem symmetry-protected topological (SSPT) states and spontaneous subsystem symmetry-breaking states associated with regular or irregular subsystem symmetries unattainable on Euclidean lattices. We design the multi-point strange correlators to detect nontrivial SSPT states and derive a sufficient condition for non-Abelian translationally invariant NUCA-generated models. Furthermore, by generalizing the NUCA to non-uniform Clifford quantum cellular automata (CQCA), we generate subsystem symmetries of the hyperbolic cluster state, extending the established correspondence between translationally invariant CQCA and subsystem symmetries. Moreover, we simulate the DP process via a probabilistic NUCA that inherits the treelike structure of the lattice, and numerically estimate percolation thresholds and the phase diagram.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a higher-order non-uniform cellular automata (NUCA) algorithm for translationally invariant regular Euclidean and hyperbolic lattices. Geometric data is incorporated via a lattice-deforming procedure into the non-uniform update rules. Applications on the hyperbolic {5,4} lattice include generation of SSPT states and spontaneous subsystem symmetry-breaking states (regular or irregular), design of multi-point strange correlators to detect nontrivial SSPT order, derivation of a sufficient condition for non-Abelian translationally invariant NUCA models, extension to non-uniform Clifford quantum cellular automata to generate subsystem symmetries of the hyperbolic cluster state, and probabilistic NUCA simulation of directed percolation to estimate thresholds and the phase diagram.

Significance. If the central claims hold, the work provides a general algorithmic framework for studying subsystem symmetries, topological phases, and percolation phenomena on hyperbolic lattices with non-Abelian translation symmetry, where uniform CA methods fail due to exponential growth. The explicit construction of strange correlators, the sufficient condition for non-Abelian models, and the CQCA extension to cluster states are concrete strengths that could enable new falsifiable predictions in quantum many-body physics on curved spaces.

major comments (2)
  1. [§3] §3 (Lattice-deforming procedure): The central claim that the procedure maps geometric data of the {5,4} lattice into non-uniform rules while exactly preserving translation invariance under the non-Abelian hyperbolic group action lacks an explicit commutator verification or orbit-stabilizer check with the translation generators. Without this, it is unclear whether the deformed rules remain group-equivariant, risking spurious correlations that would invalidate the SSPT protection and the sufficient condition for non-Abelian NUCA models.
  2. [§5] §5 (Directed percolation simulation): The numerical estimation of percolation thresholds and the phase diagram is load-bearing for the claim of broad applicability, yet the manuscript provides no system sizes, error bars, finite-size scaling analysis, or convergence diagnostics. This absence prevents assessment of whether the treelike structure inheritance yields reliable results.
minor comments (2)
  1. [§4.1] §4.1: The multi-point strange correlators are described qualitatively but would benefit from an explicit formula showing their construction and how they distinguish nontrivial SSPT states from trivial ones.
  2. Notation: The distinction between 'regular' and 'irregular' subsystem symmetries is used repeatedly but is not defined with a precise criterion or example in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We address each of the major comments below and outline the corresponding revisions.

read point-by-point responses

  1. Referee: [§3] §3 (Lattice-deforming procedure): The central claim that the procedure maps geometric data of the {5,4} lattice into non-uniform rules while exactly preserving translation invariance under the non-Abelian hyperbolic group action lacks an explicit commutator verification or orbit-stabilizer check with the translation generators. Without this, it is unclear whether the deformed rules remain group-equivariant, risking spurious correlations that would invalidate the SSPT protection and the sufficient condition for non-Abelian NUCA models.

    Authors: The lattice-deforming procedure is designed by construction to preserve the translation invariance under the non-Abelian group action, as the geometric data is incorporated in a manner that respects the lattice's symmetry generators. However, we acknowledge that an explicit verification would strengthen the presentation. In the revised version, we will include a commutator check showing that the deformed update rules commute with the translation generators, as well as an orbit-stabilizer argument to confirm group-equivariance. This addition will be placed in §3 and will not change the main results but will address the concern about potential spurious correlations. revision: yes

  2. Referee: [§5] §5 (Directed percolation simulation): The numerical estimation of percolation thresholds and the phase diagram is load-bearing for the claim of broad applicability, yet the manuscript provides no system sizes, error bars, finite-size scaling analysis, or convergence diagnostics. This absence prevents assessment of whether the treelike structure inheritance yields reliable results.

    Authors: We agree that the numerical details are important for validating the directed percolation results. The current manuscript focuses on the algorithmic framework and the qualitative phase diagram, but to improve clarity, we will add in the revised §5 the specific system sizes employed (hyperbolic lattices up to depth 12), statistical error bars from ensemble averages, finite-size scaling analysis to determine the critical thresholds, and convergence checks as a function of lattice generation. These revisions will demonstrate that the treelike inheritance provides reliable estimates. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in algorithmic construction

full rationale

The paper presents a new higher-order NUCA algorithm whose core is a lattice-deforming procedure that encodes geometric data of hyperbolic lattices into non-uniform update rules. No step reduces a claimed prediction or derived quantity to a fitted parameter or self-citation by construction; the sufficient condition for non-Abelian NUCA models, the multi-point strange correlators, and the CQCA generalization are all introduced as independent constructions rather than tautological re-labelings of inputs. The derivation chain is therefore self-contained and does not rely on any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; full text required to audit.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith.Constants phi_golden_ratio echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Solving the characteristic polynomial of the splitting matrix Eq. (25) for the {5,4} lattice, whose greatest root is the square of the golden ratio ϕ=(1+√5)/2, gives the number of nodes on each level uk=A_{2k+1}

What do these tags mean?
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supports
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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