pith. machine review for the scientific record. sign in

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

Recognition: 2 theorem links

· Lean Theorem

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

Xiang-You Huang , Jie-Yu Zhang , Peng Ye
Authors on Pith no claims yet

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
0
0 comments X

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.

pith-pipeline@v0.9.0 · 5584 in / 1192 out tokens · 36914 ms · 2026-05-14T18:10:15.015462+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

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?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
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

Works this paper leans on

118 extracted references · 6 canonical work pages · 4 internal anchors

  1. [1]

    III and Appendix D, we develop a lattice- deforming procedure for the hyperbolic lattice based on the splitting method and the language of splitting, to design the deformed lattice

    Applicability of the NUCA algorithm In Sec. III and Appendix D, we develop a lattice- deforming procedure for the hyperbolic lattice based on the splitting method and the language of splitting, to design the deformed lattice. Therefore, we first review which{p, q}lattices the splitting method and the lan- guage of the splitting can be applied to [59]. The...

  2. [2]

    Consider a quar- terQof an infinite 2d square lattice tessellated by regular rectangles, each of which is labelled as a node

    The splitting method applied to Euclidean square lattice As a simple example of how the splitting method works, we apply it to the Euclidean square lattice and construct the language of the splitting. Consider a quar- terQof an infinite 2d square lattice tessellated by regular rectangles, each of which is labelled as a node. Without loss of generality, we...

  3. [3]

    ∞X k=0 ykrk(x) + ∞X k=n ykrk(x) + n−1X k=0 yk˜rk(x) # =x −iy−j

    Consistency of NUCA(4,4)with higher-order cellular automata Next, we elaborate details and properties related to NUCA(4,4)in Sec. IIC, where we defined NUCA(4,4) directly on the square lattice. The unit translation of a site in terms of the polynomial representationxiyj is re- alized by multiplication by monomials in{x, y, x−1, y−1}. The neighboring sites...

  4. [4]

    For the{5,4}lattice, we use an alternative Fibonacci representation rather than the standard language of the splitting to simplify calcula- tion [88]

    Derivation of auxiliary functions First we provide the details of obtaining the neigh- boring relation of nodes. For the{5,4}lattice, we use an alternative Fibonacci representation rather than the standard language of the splitting to simplify calcula- tion [88]. We recall that the generating rule is→ and → in Fig. 5. On the{5,4}spanning tree, if a nodevi...

  5. [5]

    (39) commute with all the Hamil- tonian terms Eq

    Details of NUCA(5,4)-generated Hamiltonians We first verify that the NUCA(5,4)-generated sym- metry elements Eq. (39) commute with all the Hamil- tonian terms Eq. (37). We consider the nontrivial ac- tion of a Hamiltonian termZ(x iyj(1 + ¯fj · ¯y1,¯n))on the same sublattice with the symmetryX(F(x, y)) = X(P k rk(x)yk). We note that the update rule and its...

  6. [6]

    III, while we do not discuss in detail why a uniform update rule ¯fj(x)≡ ¯f(x)does not guarantee translation invariance

    Translation-invariance of NUCA-generated Hamiltonian We derive a sufficient condition for a NUCA-generated Hamiltonian to be translation-invariant in Sec. III, while we do not discuss in detail why a uniform update rule ¯fj(x)≡ ¯f(x)does not guarantee translation invariance. In this appendix, we discuss the translation invariance condition on a hyperbolic...

  7. [7]

    Y. You, T. Devakul, F. J. Burnell, and S. L. Sondhi, Subsystem symmetry protected topological order, Phys. Rev. B98, 035112 (2018)

    2018

  8. [8]

    T.Devakul, D.J.Williamson,andY.You,Classification of subsystem symmetry-protected topological phases, Phys. Rev. B98, 235121 (2018)

    2018

  9. [9]

    Devakul, Y

    T. Devakul, Y. You, F. J. Burnell, and S. L. Sondhi, Fractal Symmetric Phases of Matter, SciPost Phys.6, 007 (2019)

    2019

  10. [10]

    Devakul, Classifying local fractal subsystem symmetry-protected topological phases, Phys

    T. Devakul, Classifying local fractal subsystem symmetry-protected topological phases, Phys. Rev. B 99, 235131 (2019)

    2019

  11. [11]

    Devakul and D

    T. Devakul and D. J. Williamson, Universal quantum computation using fractal symmetry-protected cluster phases, Phys. Rev. A98, 022332 (2018)

    2018

  12. [12]

    D. T. Stephen, H. P. Nautrup, J. Bermejo-Vega, J. Eis- ert, and R. Raussendorf, Subsystem symmetries, quan- tum cellular automata, and computational phases of quantum matter, Quantum3, 142 (2019)

    2019

  13. [13]

    A. K. Daniel, R. N. Alexander, and A. Miyake, Com- putational universality of symmetry-protected topologi- callyorderedclusterphaseson2DArchimedeanlattices, Quantum4, 228 (2020)

    2020

  14. [14]

    Raussendorf and H

    R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett.86, 5188 (2001)

    2001

  15. [15]

    Zhou, M.-Y

    C. Zhou, M.-Y. Li, Z. Yan, P. Ye, and Z. Y. Meng, De- tectingsubsystemsymmetryprotectedtopologicalorder through strange correlators, Phys. Rev. B106, 214428 (2022)

    2022

  16. [16]

    Zhang, M.-Y

    J.-Y. Zhang, M.-Y. Li, and P. Ye, Higher-order cellu- lar automata generated symmetry-protected topological phases and detection through multi point strange cor- relators, PRX Quantum5, 030342 (2024)

    2024

  17. [17]

    Vijay, J

    S. Vijay, J. Haah, and L. Fu, A new kind of topological quantum order: A dimensional hierarchy of quasiparti- cles built from stationary excitations, Phys. Rev. B92, 235136 (2015)

    2015

  18. [18]

    Vijay, J

    S. Vijay, J. Haah, and L. Fu, Fracton topological order, generalized lattice gauge theory, and duality, Phys. Rev. B94, 235157 (2016)

    2016

  19. [19]

    Barkeshli, P

    M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, Symmetry fractionalization, defects, and gauging of topological phases, Phys. Rev. B100, 115147 (2019)

    2019

  20. [20]

    D. T. Stephen, J. Garre-Rubio, A. Dua, and D. J. Williamson, Subsystem symmetry enriched topological order in three dimensions, Phys. Rev. Res.2, 033331 40 (2020)

    2020

  21. [21]

    D. T. Stephen, A. Dua, J. Garre-Rubio, D. J. Williamson, and M. Hermele, Fractionalization of sub- system symmetries in two dimensions, Phys. Rev. B 106, 085104 (2022)

    2022

  22. [22]

    Fusion Rules of Mobility

    J.-Y. Zhang and P. Ye, Programmable anyon mo- bility through higher order cellular automata (2025), arXiv:2508.13961 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  23. [23]

    Asavanant, Y

    W. Asavanant, Y. Shiozawa, S. Yokoyama, B. Charoen- sombutamon, H. Emura, R. N. Alexander, S. Takeda, J. ichi Yoshikawa, N. C. Menicucci, H. Yonezawa, and A. Furusawa, Generation of time-domain-multiplexed two-dimensional cluster state, Science366, 373 (2019)

    2019

  24. [24]

    M. V. Larsen, X. Guo, C. R. Breum, J. S. Neergaard- Nielsen, and U. L. Andersen, Deterministic generation of a two-dimensional cluster state, Science366, 369 (2019)

    2019

  25. [25]

    B. P. Lanyon, P. Jurcevic, M. Zwerger, C. Hempel, E. A. Martinez, W. Dür, H. J. Briegel, R. Blatt, and C. F. Roos, Measurement-based quantum computation with trapped ions, Phys. Rev. Lett.111, 210501 (2013)

    2013

  26. [26]

    H. K. Janssen, On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing station- ary state, Z. Phys. B42, 151–154 (1981)

    1981

  27. [27]

    Grassberger, On phase transitions in schlögl’s second model, Z

    P. Grassberger, On phase transitions in schlögl’s second model, Z. Phys. B47, 365–374 (1982)

    1982

  28. [28]

    Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states, Advances in Physics49, 815 (2000)

    H. Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states, Advances in Physics49, 815 (2000)

    2000

  29. [29]

    Domany and W

    E. Domany and W. Kinzel, Equivalence of cellular au- tomata to ising models and directed percolation, Phys. Rev. Lett.53, 311 (1984)

    1984

  30. [30]

    K. A. Takeuchi, M. Kuroda, H. Chaté, and M. Sano, Directed percolation criticality in turbulent liquid crys- tals, Phys. Rev. Lett.99, 234503 (2007)

    2007

  31. [31]

    K. A. Takeuchi, M. Kuroda, H. Chaté, and M. Sano, Ex- perimental realization of directed percolation criticality in turbulent liquid crystals, Phys. Rev. E80, 051116 (2009)

    2009

  32. [32]

    Lemoult, L

    G. Lemoult, L. Shi, K. Avila, S. V. Jalikop, M. Avila, andB.Hof,Directedpercolationphasetransitiontosus- tained turbulence in couette flow, Nature Physics12, 254–258 (2016)

    2016

  33. [33]

    Lemoult, V

    G. Lemoult, V. Mukund, H.-Y. Shih, G. Linga, J. Math- iesen, N. Goldenfeld, and B. Hof, Directed percolation andpuffjammingnearthetransitiontopipeturbulence, Nature Physics20, 1339–1345 (2024)

    2024

  34. [34]

    Pizzi, A

    A. Pizzi, A. Nunnenkamp, and J. Knolle, Bistability and time crystals in long-ranged directed percolation, Nature Communications12, 1061 (2021)

    2021

  35. [35]

    Willsher, S.-W

    J. Willsher, S.-W. Liu, R. Moessner, and J. Knolle, Measurement-induced phase transition in a chaotic clas- sical many-body system, Phys. Rev. B106, 024305 (2022)

    2022

  36. [36]

    R. C. Brower, C. V. Cogburn, A. L. Fitzpatrick, D. Howarth, and C.-I. Tan, Lattice setup for quantum field theory inads2, Phys. Rev. D103, 094507 (2021)

    2021

  37. [37]

    R. C. Brower, C. V. Cogburn, and E. Owen, Hyperbolic lattice for scalar field theory inads3, Phys. Rev. D105, 114503 (2022)

    2022

  38. [38]

    Boettcher, A

    I. Boettcher, A. V. Gorshkov, A. J. Kollár, J. Maciejko, S. Rayan, and R. Thomale, Crystallography of hyper- bolic lattices, Phys. Rev. B105, 125118 (2022)

    2022

  39. [39]

    F.R.LuxandE.Prodan,Convergingperiodicboundary conditions and detection of topological gaps on regular hyperbolic tessellations, Phys. Rev. Lett.131, 176603 (2023)

    2023

  40. [40]

    Maciejko and S

    J. Maciejko and S. Rayan, Hyperbolic band theory, Sci- ence Advances7, eabe9170 (2021)

    2021

  41. [41]

    Maciejko and S

    J. Maciejko and S. Rayan, Automorphic bloch theo- rems for hyperbolic lattices, Proceedings of the National Academy of Sciences119, e2116869119 (2022)

    2022

  42. [42]

    P. M. Lenggenhager, J. Maciejko, and T. c. v. Bzdušek, Non-abelian hyperbolic band theory from supercells, Phys. Rev. Lett.131, 226401 (2023)

    2023

  43. [43]

    A. J. Kollár, M. Fitzpatrick, and A. A. Houck, Hyper- bolic lattices in circuit quantum electrodynamics, Na- ture571, 45–50 (2019)

    2019

  44. [44]

    Helbig, A

    P.M.Lenggenhager, A.Stegmaier, L.K.Upreti, T.Hof- mann, T. Helbig, A. Vollhardt, M. Greiter, C. H. Lee, S. Imhof, H. Brand, T. Kießling, I. Boettcher, T. Neu- pert, R. Thomale, and T. Bzdušek, Simulating hyper- bolic space on a circuit board, Nature Communications 13, 4373 (2022)

    2022

  45. [45]

    Huang, L

    L. Huang, L. He, W. Zhang, H. Zhang, D. Liu, X. Feng, F. Liu, K. Cui, Y. Huang, W. Zhang, and X. Zhang, Hy- perbolic photonic topological insulators, Nature Com- munications15, 1647 (2024)

    2024

  46. [46]

    A. Chen, Y. Guan, P. M. Lenggenhager, J. Maciejko, I. Boettcher, and T. c. v. Bzdušek, Symmetry and topol- ogy of hyperbolic haldane models, Phys. Rev. B108, 085114 (2023)

    2023

  47. [47]

    Tummuru, A

    T. Tummuru, A. Chen, P. M. Lenggenhager, T. Ne- upert, J. Maciejko, and T. c. v. Bzdušek, Hyperbolic non-abelian semimetal, Phys. Rev. Lett.132, 206601 (2024)

    2024

  48. [48]

    A. Chen, J. Maciejko, and I. Boettcher, Anderson lo- calization transition in disordered hyperbolic lattices, Phys. Rev. Lett.133, 066101 (2024)

    2024

  49. [49]

    Huang, Y

    X.-Y. Huang, Y. Zhou, and P. Ye, Entanglement scaling behaviors of free fermions on hyperbolic lattices, Phys. Rev. Res.7, 023098 (2025)

    2025

  50. [50]

    P. M. Lenggenhager, S. Dey, T. c. v. Bzdušek, and J. Maciejko, Hyperbolic spin liquids, Phys. Rev. Lett. 135, 076604 (2025)

    2025

  51. [51]

    Dutkiewicz, M

    K. Dutkiewicz, M. Płodzień, A. Rojo-Francàs, B. Juliá- Díaz, M. Lewenstein, and T. Grass, Bose-einstein con- densation in exotic lattice geometries, Phys. Rev. A 113, 013320 (2026)

    2026

  52. [52]

    Saraidaris and A

    D. Saraidaris and A. Jahn, Critical spin models from holographic disorder, Quantum9, 1808 (2025)

    2025

  53. [53]

    Yan, Hyperbolic fracton model, subsystem symme- try, and holography, Phys

    H. Yan, Hyperbolic fracton model, subsystem symme- try, and holography, Phys. Rev. B99, 155126 (2019)

    2019

  54. [54]

    Yan, Hyperbolic fracton model, subsystem symme- try, and holography

    H. Yan, Hyperbolic fracton model, subsystem symme- try, and holography. ii. the dual eight-vertex model, Phys. Rev. B100, 245138 (2019)

    2019

  55. [55]

    Hyperbolic Fracton Model, Subsystem Symmetry and Holography III: Extension to Generic Tessellations

    Y. Shokeeb, L. D. C. Jaubert, and H. Yan, Hyper- bolic fracton model, subsystem symmetry and holog- raphy iii: Extension to generic tessellations (2026), arXiv:2510.25994 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  56. [56]

    N. P. Breuckmann and B. M. Terhal, Constructions and noisethresholdofhyperbolicsurfacecodes,IEEETrans- actions on Information Theory62, 3731 (2016)

    2016

  57. [57]

    Fahimniya, H

    A. Fahimniya, H. Dehghani, K. Bharti, S. Mathew, A. J. Kollár, A. V. Gorshkov, and M. J. Gullans, Fault- tolerant hyperbolic Floquet quantum error correcting codes, Quantum9, 1849 (2025)

    2025

  58. [58]

    A. A. Mahmoud, K. M. Ali, and S. Rayan, Systematic 41 approach to hyperbolic quantum error correction codes, Phys. Rev. A113, 042426 (2026)

    2026

  59. [59]

    Asaduzzaman, S

    M. Asaduzzaman, S. Catterall, J. Hubisz, R. Nelson, and J. Unmuth-Yockey, Holography on tessellations of hyperbolic space, Phys. Rev. D102, 034511 (2020)

    2020

  60. [60]

    A. Jahn, Z. Zimborás, and J. Eisert, Central charges of aperiodic holographic tensor-network models, Phys. Rev. A102, 042407 (2020)

    2020

  61. [61]

    Boyle, M

    L. Boyle, M. Dickens, and F. Flicker, Conformal qua- sicrystals and holography, Phys. Rev. X10, 011009 (2020)

    2020

  62. [62]

    Boyle and J

    L. Boyle and J. Kulp, Holographic foliations: Self- similar quasicrystals from hyperbolic honeycombs, Phys. Rev. D111, 046001 (2025)

    2025

  63. [63]

    S. Dey, A. Chen, P. Basteiro, A. Fritzsche, M. Gre- iter, M. Kaminski, P. M. Lenggenhager, R. Meyer, R. Sorbello, A. Stegmaier, R. Thomale, J. Erdmenger, andI.Boettcher,Simulatingholographicconformalfield theories on hyperbolic lattices, Phys. Rev. Lett.133, 061603 (2024)

    2024

  64. [64]

    Mertens and C

    S. Mertens and C. Moore, Percolation thresholds in hy- perbolic lattices, Phys. Rev. E96, 042116 (2017)

    2017

  65. [65]

    Margenstern, Cellular automata in hyperbolic spaces, inCellular Automata: A Volume in the Encyclo- pedia of Complexity and Systems Science, Second Edi- tion, edited by A

    M. Margenstern, Cellular automata in hyperbolic spaces, inCellular Automata: A Volume in the Encyclo- pedia of Complexity and Systems Science, Second Edi- tion, edited by A. Adamatzky (Springer US, New York, NY, 2018) pp. 11–27

    2018

  66. [66]

    Schrauth, Y

    M. Schrauth, Y. Thurn, F. Goth, J. S. E. Portela, D. Herdt, and F. Dusel, HYPERTILING — a high per- formance Python library for the generation and visual- ization of hyperbolic lattices, SciPost Phys. Codebases , 34 (2024)

    2024

  67. [67]

    Wolfram, Statistical mechanics of cellular automata, Rev

    S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys.55, 601 (1983)

    1983

  68. [68]

    Wolfram, Cellular automata as models of complexity, Nature (London)311, 419 (1984)

    S. Wolfram, Cellular automata as models of complexity, Nature (London)311, 419 (1984)

    1984

  69. [69]

    T.Toffoli,Computationandconstructionuniversalityof reversible cellular automata, Journal of Computer and System Sciences15, 213 (1977)

    1977

  70. [70]

    M.Rollier, K.M.Zielinski, A.J.Daly, O.M.Bruno,and J. M. Baetens, A comprehensive taxonomy of cellular automata, Communications in Nonlinear Science and Numerical Simulation140, 108362 (2025)

    2025

  71. [71]

    Louis and F

    P.-Y. Louis and F. R. Nardi, eds.,Probabilistic Cellular Automata: Theory, Applications and Future Perspec- tives(Springer International Publishing, 2018)

    2018

  72. [72]

    Biswas, Y

    S. Biswas, Y. H. Kwan, and S. A. Parameswaran, Be- yond the freshman’s dream: Classical fractal spin liq- uids from matrix cellular automata in three-dimensional lattice models, Phys. Rev. B105, 224410 (2022)

    2022

  73. [73]

    Raussendorf, Quantum cellular automaton for uni- versal quantum computation, Phys

    R. Raussendorf, Quantum cellular automaton for uni- versal quantum computation, Phys. Rev. A72, 022301 (2005)

    2005

  74. [74]

    T. J. Osborne, Efficient approximation of the dynamics of one-dimensional quantum spin systems, Phys. Rev. Lett.97, 157202 (2006)

    2006

  75. [75]

    Farrelly, A review of Quantum Cellular Automata, Quantum4, 368 (2020)

    T. Farrelly, A review of Quantum Cellular Automata, Quantum4, 368 (2020)

    2020

  76. [76]

    Arrighi, An overview of quantum cellular automata, Natural Computing18, 885–899 (2019)

    P. Arrighi, An overview of quantum cellular automata, Natural Computing18, 885–899 (2019)

    2019

  77. [77]

    Pries, Thanailakis, and Card, Group properties of cellu- lar automata and vlsi applications, IEEE Transactions on ComputersC-35, 1013 (1986)

    1986

  78. [78]

    Fuks-acute, Solution of the density classification problem with two cellular automata rules, Phys

    H. Fuks-acute, Solution of the density classification problem with two cellular automata rules, Phys. Rev. E55, R2081 (1997)

    1997

  79. [79]

    Ghosh and S

    S. Ghosh and S. Basu, Evolution patterns of some one-dimensional non-uniform cellular automata, inPro- ceedings of First Asian Symposium on Cellular Au- tomata Technology, edited by S. Das and G. J. Mar- tinez (Springer Nature Singapore, Singapore, 2022) pp. 69–77

    2022

  80. [80]

    Paul and S

    S. Paul and S. Das, Temporally non-uniform cellular automata: Surjectivity, reversibility and cyclic behavior (2026), arXiv:2411.17421 [cs.FL]

    work page arXiv 2026

Showing first 80 references.