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arXiv:2607.15326

Quantum error-correcting codes from aperiodic monotiles: the Hat and the Spectre

Josep Batle, Adam Bednorz

quant-phmath.MGSubmitted Jul 16, 2026• Announced Jul 17, 2026• 0 citations
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Abstract

Li and Boyle showed that the Penrose tiling defines a quantum error-correcting code: superpositions of tilings over isometry orbits protect quantum information against erasure of any bounded region. We extend the construction to the aperiodic monotiles discovered by Smith, Myers, Kaplan and Goodman-Strauss. For the Hat, we prove strong local indistinguishability for all Hat tilings, and we prove local recoverability unconditionally for all nonsingular Hat tilings via the torus parametrization of the underlying cut-and-project scheme. The remaining singular case reduces to one sharply posed geometric question -- can a region that is a union of hats be retiled a second way? -- which we verify computationally has no counterexample up to a substantial scale: a certified $2490$-tile patch admits precisely one tiling by hats, so all $2^{2490}$ of its tile-subregions retile uniquely. Unlike the Penrose, Ammann-Beenker and Fibonacci tilings, both monotiles form two local-indistinguishability classes, so their code spaces split into two erasure-correcting sectors carrying a superselected classical label. Whether the label survives depends on which isometries are gauged: the Spectre's classes are exchanged by a $30^{\circ}$ rotation and merge once all proper isometries are gauged, whereas the Hat's are exchanged only by reflections. Under the physically natural gauge group $SE(2)$, the Hat code therefore stores one robust classical bit -- the handedness of its long-range order, readable in any bounded window with separation $Δ= |K|\sqrt{5}/3$ -- alongside its protected quantum sectors. It is the reflexible monotile, not the chiral one, that carries the chirality bit. We give the exact Perron-Frobenius data of both codes, including the per-class reflected-Hat frequencies $(3\mp\sqrt{5})/6$ and the Spectre orientation-class frequencies $(5\pm\sqrt{15})/10$.

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The Penrose Tiling is a Quantum Error-Correcting Code

Zhi Li, Latham A. Boyle

The Penrose tiling (PT) is an intrinsically non-periodic way of tiling the plane, with many remarkable properties. A quantum error-correcting code (QECC) is a clever way of protecting quantum information from noise, by encoding the information with a sophisticated type of redundancy. Although PTs and QECCs might seem completely unrelated, in this paper we point out that PTs give rise to (or, in a sense, are) a remarkable new type of QECC. In this code, quantum information is encoded through quantum geometry, and any local errors or erasures in any finite region, no matter how large, may be diagnosed and corrected. We also construct variants of this code (based on the Ammann-Beenker and Fibonacci tilings) that can live on finite spatial tori, in discrete spin systems, or in an arbitrary number of spatial dimensions. We discuss connections to quantum computing, condensed matter physics, and quantum gravity.

arXiv:2311.13040•Nov 21, 2023•0 citations

Proof of a positive coherent-error threshold for topological quantum codes

Shiro Tamiya, Masato Koashi

Threshold analyses of quantum error-correcting codes are well established for stochastic error models, in which errors occur randomly with given probabilities. However, errors in actual devices can also be coherent, such as unwanted $Z$ rotations due to imperfect control, which are not captured by stochastic error models. For the surface code, numerical studies have indicated threshold behavior even under coherent errors, but a rigorous proof of threshold existence is lacking. Here we prove that a positive threshold for coherent $Z$-rotation errors exists for quantum low-density parity-check codes with a bounded number of logical qubits, including the surface code and other topological codes. Specifically, we show that the maximum-likelihood Pauli recovery suppresses the entanglement infidelity exponentially in the code distance up to a prefactor linear in the number of physical qubits whenever the rotation angles lie below a constant that is independent of the code size. The proof combines Fourier analysis to retain the interference among the amplitudes of coherent errors with the cluster expansion of abstract polymer models. Our results expand the theoretical foundation of quantum error correction and offer a statistical-mechanical description of quantum error correction beyond stochastic errors.

quant-pharXiv:2609.20708•Sep 18, 2026•0 citations

From Bounded Surjections to Quantum Error-Correcting Codes: An Operator-Algebraic Construction

Sudoma, Oksana

Abstract We establish that every bounded surjection π : B ↠ H between Banach and Hilbertspaces canonically induces a quantum error-correcting code structure. The kernel ker π capturesunobservable degrees of freedom, while the quotient B/ ker π encodes protected logical information.This perspective: (1) unifies subspace, subsystem, and operator-algebraic codes under oneframework; (2) provides functorial code transformations via morphisms of surjections; (3)enables computational exploration through kernel geometry analysis; (4) extends naturally toinfinite dimensions via GNS construction without requiring infinite stabilizer sets. We prove thecorrespondence is functorial with a universal property, demonstrate stability under perturbations,and show the Knill-Laflamme conditions emerge from kernel structure. Examples include theShor code arising from syndrome extraction, holographic codes from bulk-boundary maps, andoperator-algebraic codes from GNS representations. Computational validation (ExperimentsE49, E50) confirms kernel dimension predictions with machine-precision accuracy and revealsrank transition behavior in continuous code families, with dimension derivatives |∆k/∆t| ∼ 102at transition points. While finite-dimensional codes are mathematically equivalent to knownconstructions, our framework provides conceptual unification: (1) kernel geometry explains whydifferent QEC approaches converge on the same codes; (2) continuous code families handle ranktransitions without stabilizer redefinition; (3) infinite dimensions via GNS construction avoidinfinite stabilizer sets; (4) functorial transformations arise naturally from surjection morphisms;(5) the approach aligns with physical intuition—error correction protects exactly the informationthat survives observation. Where Surjections Appear Naturally Quantum error correction typically starts with a codespace—a carefully chosen subspace of a larger Hilbert space [ KL97, NC10 , Got97 ]. 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Some bulk configurations are invisible from the boundary—they lie in the kernelof the bulk-to-boundary map. The observable bulk information forms a quotient space. Could thisquotient structure automatically define error correction? Our Contribution We prove that every bounded surjection π : B ↠ H from a Banach space ontoa Hilbert space canonically induces a quantum error-correcting code via the quotient constructionfrom functional analysis [Rud91, Con90]. The construction is:1. Universal: Works for finite and infinite-dimensional systems2. Natural: QEC emerges from observability structure (kernel geometry)3. Functorial: Morphisms between surjections induce code transformations Positioning in the Literature This work occupies a unique position at the intersection of threeapproaches to quantum error correction:• vs. Stabilizer QEC [Got97 ]: We provide a dual perspective where codes emerge fromobservables (what we can measure) rather than symmetries (what commutes). This isconceptually inverse but mathematically equivalent in finite dimensions.• vs. Operator Algebra QEC [ KLP05, Pou05 , KL24]: While OAQEC uses von Neumannalgebras to generalize stabilizers, we use surjections to generalize projections. Our frameworkencompasses OAQEC as the special case where B carries algebra structure (Proposition 4).• vs. Holographic QEC [ ADH15, PYHP15]: Holographic codes implicitly use bulk-to-boundary maps. We make this explicit, showing these maps are surjections whose kernelsencode gauge redundancy. Our Contribution We do not claim new codes but rather a unifying framework that:1. Explains why these approaches converge on the same codes (they all induce surjections)2. Provides computational tools via kernel geometry analysis3. Enables systematic code exploration by designing surjections rather than searching for stabi-lizers4. Naturally extends to infinite dimensions where stabilizers become unwieldyThe key insight: error correction is fundamentally about preserving observable information underlimited measurement resolution—the kernel captures what’s unmeasurable, the quotient preserveswhat survives. Experiments E49: Kernel Geometry Visualization Validates [[4,2,2]] quantum code construction from bounded surjections. Key results: Code distance d=2 verified Kernel orthogonality: 10^-16 (machine precision) Logical operators detected Runtime: < 1 second Documentation: See experiments/E49_kernel_geometry_viz/ABOUT_THIS_EXPERIMENT.md for complete research narrative (hypothesis → results → interpretation) on https://github.com/boonespacedog/surjection-to-qec.git . Quick start: cd experiments/E49_kernel_geometry_viz python3 -m venv venv source venv/bin/activate pip install -r requirements.txt python3 main.py E50: Functorial Code Morphisms Continuous code family evolution with rank transition detection. 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•Nov 15, 2025•0 citations

Silent Corruption in Aperiodic Substrates: A Relational Integrity Diagnostic for Error-Correcting Architectures

K. T. Niedzwiecki

Li and Boyle (2024) demonstrated that aperiodic tilings — including Penrose,Ammann-Beenker (AB), and Fibonacci geometries — give rise to a remarkable class ofquantum error-correcting codes (QECCs) in which quantum information is encoded throughthe geometric structure of the tiling itself. Their analysis establishes that these codes arerobust against erasure errors, where a finite patch of the tiling is deleted and reconstructedfrom the surrounding geometry. Here we introduce a substrate-level diagnostic for a distinctperturbation class we term silent relational corruption: the scrambling of a graph'sconnectivity topology while preserving local degree statistics. Unlike erasure, silent relationalcorruption produces no detectable gap in the substrate — the perturbation leaves no visibletrace. Although motivated by aperiodic quantum error-correcting codes, the present audit isclassical: it evaluates recoverability of privileged-persistence sites under controlled graphperturbations, not the performance of a quantum code. Using a matched degree-preservingrewiring protocol applied to AB (N=273) and Penrose (N=7,160) substrates generated viacut-and-project methods, we demonstrate that vulnerability to silent relational corruption issubstrate-dependent and predictable from a geometric invariant: the exo/endo persistenceaxis. AB substrates, whose structural logic is stored in hidden perpendicular-spacecoordinates (exo-addressed), retain near-ceiling identity recoverability under full relationalscrambling (AUC = 1.000). Penrose substrates, whose structural logic resides entirely in thelocal relational weave (endo-reconstructive), show substantially reduced identityrecoverability (AUC = 0.800). We further demonstrate that identity persistence and functionalreconstruction are operationally distinct processes — the former is address-led, the latterweave-led — and that hybrid architectures incorporating both channels providecomplementary redundancy under mixed damage. These findings suggest that silent relationalcorruption resistance should be evaluated as a substrate selection criterion for aperiodicerror-correcting architectures. Contact: niedzkatie@gmail.com

•May 27, 2026•0 citations

A diagrammatic field theory of quantum error correction

Steven Rayan

We develop a field-theoretic framework for quantum error correction centred on fusion-space codes in unitary fusion categories. Admissible clusters determine total-charge sectors and orthogonal footprint projectors recording locally visible data left by error histories. The central distinction is between diagnostic footprint algebras and syndrome-admissible commuting algebras: the latter can be measured without revealing logical information and resolve chosen error representatives into measured sectors. For such algebras, exact correctability is equivalent to fibrewise Knill--Laflamme conditions, yielding a measure-then-recover factorization. Under a contractible-vacuum locality hypothesis, closed neutral composites give a categorical sufficient criterion for scalar action on the code. In the Ising theory, four $σ$ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-$σ$ code instead admits a syndrome-admissible pair-charge measurement and exact recovery from an explicit Majorana bilinear error. A second bilinear has the same measured footprint but differs by a logical bit flip, producing a concrete nontrivial footprint fibre and genuine decoding ambiguity. We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights. For growing code families, we prove a conditional Peierls-type threshold theorem: bounded connected-region growth, local stochastic noise, local neutralizability of small residual components, and componentwise decoder balance imply $\Pr_L(\mathrm{fail})\le C|Ω_L|e^{-cL}$ below a nonzero constant error rate. We conclude with representation-theoretic and algebro-geometric directions involving tube and Hopf algebras, Yangian-type structures, Higgs bundles, spectral curves, Jacobians, and abelian varieties.

quant-pharXiv:2607.08911•Jul 13, 2026•0 citations

Constrained models in aperiodic systems

Singh, Shobhna

Constrained models play a pivotal role in condensed matter physics by capturing essential features of strongly interacting many-body systems. These models impose local or global restrictions that mirror real physical constraints, enabling the study of complex emergent behaviors. For instance, hard-core dimer models reflect physical systems with strong local constraints due to repulsive interactions. Similarly, loop models, vertex models, and height representations provide constrained frameworks that reveal rich phase structures and critical phenomena. In quantum systems, constraints often emerge naturally from microscopic Hamiltonian and lead to phenomena such as fractionalization and quantum spin liquids. Constrained models act as fertile grounds for theoretical insight, enabling exact solutions, mapping to field theories, and exploring universal behaviors. They also bridge condensed matter with disciplines like statistical mechanics, combinatorics, and computational complexity, offering a unified language to study order, disorder, and transitions in systems ranging from magnets to artificial spin ices and cold atomic gases. This thesis investigates critical phenomena, strongly correlated models, and optimization problems on aperiodic lattices, with a particular focus on NP-hard problems, dimer models, and their classical and quantum variants. Moving beyond periodic lattices like square and honeycomb structures, this research delves into the less understood area of aperiodic systems, such as quasicrystals, which exhibit long-range order without translational symmetry. These structures challenge traditional paradigms of condensed matter physics and open avenues for discovering new critical behaviors. I present original contributions that combine analytical methods and numerical algorithms to study constrained systems on quasicrystalline tilings. These include the solution of NP-hard optimization problems like the Hamiltonian cycle on Ammann-Beenker tilings, an energy-based worm algorithm for simulating classical dimers on random graphs and modified Penrose tilings, and the analytical solution of the dimer partition function on the newly discovered aperiodic Spectre tiling. I further extend the study to quantum systems by analytically investigating the quantum dimer model (QDM) on the Spectre tiling and numerically analyzing it on the Ammann-Beenker tiling, revealing distinctive ground-state properties and quantum phase transitions that arise due to aperiodicity and local tiling constraints. This work builds interdisciplinary bridges between condensed matter physics, graph theory, and computational complexity. It also sheds light on the feasibility and limitations of transferring universal results from periodic systems to aperiodic ones.

•Sep 1, 2025•0 citations
Li and Boyle showed that the Penrose tiling defines a quantum error-correcting code: superpositions of tilings over isometry orbits protect quantum information against erasure of any bounded region. We extend the construction to the aperiodic monotiles discovered by Smith, Myers, Kaplan and Goodman-Strauss. For the Hat, we prove strong local indistinguishability for all Hat tilings, and we prove local recoverability unconditionally for all nonsingular Hat tilings via the torus parametrization of the underlying cut-and-project scheme. The remaining singular case reduces to one sharply posed geometric question -- can a region that is a union of hats be retiled a second way? -- which we verify computationally has no counterexample up to a substantial scale: a certified 249024902490-tile patch admits precisely one tiling by hats, so all 224902^{2490}22490 of its tile-subregions retile uniquely. Unlike the Penrose, Ammann-Beenker and Fibonacci tilings, both monotiles form two local-indistinguishability classes, so their code spaces split into two erasure-correcting sectors carrying a superselected classical label. Whether the label survives depends on which isometries are gauged: the Spectre's classes are exchanged by a 30∘30^{\circ}30∘ rotation and merge once all proper isometries are gauged, whereas the Hat's are exchanged only by reflections. Under the physically natural gauge group SE(2)SE(2)SE(2), the Hat code therefore stores one robust classical bit -- the handedness of its long-range order, readable in any bounded window with separation Δ=∣K∣5/3Δ= |K|\sqrt{5}/3Δ=∣K∣5​/3 -- alongside its protected quantum sectors. It is the reflexible monotile, not the chiral one, that carries the chirality bit. We give the exact Perron-Frobenius data of both codes, including the per-class reflected-Hat frequencies (3∓5)/6(3\mp\sqrt{5})/6(3∓5​)/6 and the Spectre orientation-class frequencies (5±15)/10(5\pm\sqrt{15})/10(5±15​)/10.
The Penrose tiling (PT) is an intrinsically non-periodic way of tiling the plane, with many remarkable properties. A quantum error-correcting code (QECC) is a clever way of protecting quantum information from noise, by encoding the information with a sophisticated type of redundancy. Although PTs and QECCs might seem completely unrelated, in this paper we point out that PTs give rise to (or, in a sense, are) a remarkable new type of QECC. In this code, quantum information is encoded through quantum geometry, and any local errors or erasures in any finite region, no matter how large, may be diagnosed and corrected. We also construct variants of this code (based on the Ammann-Beenker and Fibonacci tilings) that can live on finite spatial tori, in discrete spin systems, or in an arbitrary number of spatial dimensions. We discuss connections to quantum computing, condensed matter physics, and quantum gravity.
Threshold analyses of quantum error-correcting codes are well established for stochastic error models, in which errors occur randomly with given probabilities. However, errors in actual devices can also be coherent, such as unwanted ZZZ rotations due to imperfect control, which are not captured by stochastic error models. For the surface code, numerical studies have indicated threshold behavior even under coherent errors, but a rigorous proof of threshold existence is lacking. Here we prove that a positive threshold for coherent ZZZ-rotation errors exists for quantum low-density parity-check codes with a bounded number of logical qubits, including the surface code and other topological codes. Specifically, we show that the maximum-likelihood Pauli recovery suppresses the entanglement infidelity exponentially in the code distance up to a prefactor linear in the number of physical qubits whenever the rotation angles lie below a constant that is independent of the code size. The proof combines Fourier analysis to retain the interference among the amplitudes of coherent errors with the cluster expansion of abstract polymer models. Our results expand the theoretical foundation of quantum error correction and offer a statistical-mechanical description of quantum error correction beyond stochastic errors.
Abstract We establish that every bounded surjection π : B ↠ H between Banach and Hilbertspaces canonically induces a quantum error-correcting code structure. The kernel ker π capturesunobservable degrees of freedom, while the quotient B/ ker π encodes protected logical information.This perspective: (1) unifies subspace, subsystem, and operator-algebraic codes under oneframework; (2) provides functorial code transformations via morphisms of surjections; (3)enables computational exploration through kernel geometry analysis; (4) extends naturally toinfinite dimensions via GNS construction without requiring infinite stabilizer sets. We prove thecorrespondence is functorial with a universal property, demonstrate stability under perturbations,and show the Knill-Laflamme conditions emerge from kernel structure. Examples include theShor code arising from syndrome extraction, holographic codes from bulk-boundary maps, andoperator-algebraic codes from GNS representations. Computational validation (ExperimentsE49, E50) confirms kernel dimension predictions with machine-precision accuracy and revealsrank transition behavior in continuous code families, with dimension derivatives |∆k/∆t| ∼ 102at transition points. While finite-dimensional codes are mathematically equivalent to knownconstructions, our framework provides conceptual unification: (1) kernel geometry explains whydifferent QEC approaches converge on the same codes; (2) continuous code families handle ranktransitions without stabilizer redefinition; (3) infinite dimensions via GNS construction avoidinfinite stabilizer sets; (4) functorial transformations arise naturally from surjection morphisms;(5) the approach aligns with physical intuition—error correction protects exactly the informationthat survives observation. Where Surjections Appear Naturally Quantum error correction typically starts with a codespace—a carefully chosen subspace of a larger Hilbert space [ KL97, NC10 , Got97 ]. But nature oftenpresents us with projection maps first:• Coarse-graining: Mapping microscopic states to macroscopic observables• Symmetry reduction: Projecting onto gauge-invariant subspaces• Partial trace: Observing subsystems of composite quantum states• Boundary limits: In AdS/CFT, mapping bulk fields to boundary observablesEach scenario involves a map from a larger space to a smaller observable space. Informationthat survives this projection is precisely what remains accessible. The question naturally arises:Can these projections automatically induce quantum error-correcting structure? This paper answers affirmatively: every bounded surjection canonically induces quantumerror correction structure. The construction applies broadly—from finite qubit codes to infinite-dimensional operator algebras. Consider the concrete scenario of holographic duality: bulk quantum fields map to boundaryCFT observables. Some bulk configurations are invisible from the boundary—they lie in the kernelof the bulk-to-boundary map. The observable bulk information forms a quotient space. Could thisquotient structure automatically define error correction? Our Contribution We prove that every bounded surjection π : B ↠ H from a Banach space ontoa Hilbert space canonically induces a quantum error-correcting code via the quotient constructionfrom functional analysis [Rud91, Con90]. The construction is:1. Universal: Works for finite and infinite-dimensional systems2. Natural: QEC emerges from observability structure (kernel geometry)3. Functorial: Morphisms between surjections induce code transformations Positioning in the Literature This work occupies a unique position at the intersection of threeapproaches to quantum error correction:• vs. Stabilizer QEC [Got97 ]: We provide a dual perspective where codes emerge fromobservables (what we can measure) rather than symmetries (what commutes). This isconceptually inverse but mathematically equivalent in finite dimensions.• vs. Operator Algebra QEC [ KLP05, Pou05 , KL24]: While OAQEC uses von Neumannalgebras to generalize stabilizers, we use surjections to generalize projections. Our frameworkencompasses OAQEC as the special case where B carries algebra structure (Proposition 4).• vs. Holographic QEC [ ADH15, PYHP15]: Holographic codes implicitly use bulk-to-boundary maps. We make this explicit, showing these maps are surjections whose kernelsencode gauge redundancy. Our Contribution We do not claim new codes but rather a unifying framework that:1. Explains why these approaches converge on the same codes (they all induce surjections)2. Provides computational tools via kernel geometry analysis3. Enables systematic code exploration by designing surjections rather than searching for stabi-lizers4. Naturally extends to infinite dimensions where stabilizers become unwieldyThe key insight: error correction is fundamentally about preserving observable information underlimited measurement resolution—the kernel captures what’s unmeasurable, the quotient preserveswhat survives. Experiments E49: Kernel Geometry Visualization Validates [[4,2,2]] quantum code construction from bounded surjections. Key results: Code distance d=2 verified Kernel orthogonality: 10^-16 (machine precision) Logical operators detected Runtime: < 1 second Documentation: See experiments/E49_kernel_geometry_viz/ABOUT_THIS_EXPERIMENT.md for complete research narrative (hypothesis → results → interpretation) on https://github.com/boonespacedog/surjection-to-qec.git . Quick start: cd experiments/E49_kernel_geometry_viz python3 -m venv venv source venv/bin/activate pip install -r requirements.txt python3 main.py E50: Functorial Code Morphisms Continuous code family evolution with rank transition detection. Key results: Rank transitions detected at t = 2^n - 1 (discrete phase transitions!) Dimension jumps: |Δdim/Δt| = 198 at critical points Topological protection discovered (codes resist smooth deformation) Runtime: ~1.5 seconds Documentation: See experiments/E50_functorial_code_morphisms/ABOUT_THIS_EXPERIMENT.md for complete research narrative and link to topological interpretations on https://github.com/boonespacedog/surjection-to-qec.git .
Li and Boyle (2024) demonstrated that aperiodic tilings — including Penrose,Ammann-Beenker (AB), and Fibonacci geometries — give rise to a remarkable class ofquantum error-correcting codes (QECCs) in which quantum information is encoded throughthe geometric structure of the tiling itself. Their analysis establishes that these codes arerobust against erasure errors, where a finite patch of the tiling is deleted and reconstructedfrom the surrounding geometry. Here we introduce a substrate-level diagnostic for a distinctperturbation class we term silent relational corruption: the scrambling of a graph'sconnectivity topology while preserving local degree statistics. Unlike erasure, silent relationalcorruption produces no detectable gap in the substrate — the perturbation leaves no visibletrace. Although motivated by aperiodic quantum error-correcting codes, the present audit isclassical: it evaluates recoverability of privileged-persistence sites under controlled graphperturbations, not the performance of a quantum code. Using a matched degree-preservingrewiring protocol applied to AB (N=273) and Penrose (N=7,160) substrates generated viacut-and-project methods, we demonstrate that vulnerability to silent relational corruption issubstrate-dependent and predictable from a geometric invariant: the exo/endo persistenceaxis. AB substrates, whose structural logic is stored in hidden perpendicular-spacecoordinates (exo-addressed), retain near-ceiling identity recoverability under full relationalscrambling (AUC = 1.000). Penrose substrates, whose structural logic resides entirely in thelocal relational weave (endo-reconstructive), show substantially reduced identityrecoverability (AUC = 0.800). We further demonstrate that identity persistence and functionalreconstruction are operationally distinct processes — the former is address-led, the latterweave-led — and that hybrid architectures incorporating both channels providecomplementary redundancy under mixed damage. These findings suggest that silent relationalcorruption resistance should be evaluated as a substrate selection criterion for aperiodicerror-correcting architectures. Contact: niedzkatie@gmail.com
We develop a field-theoretic framework for quantum error correction centred on fusion-space codes in unitary fusion categories. Admissible clusters determine total-charge sectors and orthogonal footprint projectors recording locally visible data left by error histories. The central distinction is between diagnostic footprint algebras and syndrome-admissible commuting algebras: the latter can be measured without revealing logical information and resolve chosen error representatives into measured sectors. For such algebras, exact correctability is equivalent to fibrewise Knill--Laflamme conditions, yielding a measure-then-recover factorization. Under a contractible-vacuum locality hypothesis, closed neutral composites give a categorical sufficient criterion for scalar action on the code. In the Ising theory, four σσσ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-σσσ code instead admits a syndrome-admissible pair-charge measurement and exact recovery from an explicit Majorana bilinear error. A second bilinear has the same measured footprint but differs by a logical bit flip, producing a concrete nontrivial footprint fibre and genuine decoding ambiguity. We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights. For growing code families, we prove a conditional Peierls-type threshold theorem: bounded connected-region growth, local stochastic noise, local neutralizability of small residual components, and componentwise decoder balance imply Pr⁡L(fail)≤C∣ΩL∣e−cL\Pr_L(\mathrm{fail})\le C|Ω_L|e^{-cL}PrL​(fail)≤C∣ΩL​∣e−cL below a nonzero constant error rate. We conclude with representation-theoretic and algebro-geometric directions involving tube and Hopf algebras, Yangian-type structures, Higgs bundles, spectral curves, Jacobians, and abelian varieties.
Constrained models play a pivotal role in condensed matter physics by capturing essential features of strongly interacting many-body systems. These models impose local or global restrictions that mirror real physical constraints, enabling the study of complex emergent behaviors. For instance, hard-core dimer models reflect physical systems with strong local constraints due to repulsive interactions. Similarly, loop models, vertex models, and height representations provide constrained frameworks that reveal rich phase structures and critical phenomena. In quantum systems, constraints often emerge naturally from microscopic Hamiltonian and lead to phenomena such as fractionalization and quantum spin liquids. Constrained models act as fertile grounds for theoretical insight, enabling exact solutions, mapping to field theories, and exploring universal behaviors. They also bridge condensed matter with disciplines like statistical mechanics, combinatorics, and computational complexity, offering a unified language to study order, disorder, and transitions in systems ranging from magnets to artificial spin ices and cold atomic gases. This thesis investigates critical phenomena, strongly correlated models, and optimization problems on aperiodic lattices, with a particular focus on NP-hard problems, dimer models, and their classical and quantum variants. Moving beyond periodic lattices like square and honeycomb structures, this research delves into the less understood area of aperiodic systems, such as quasicrystals, which exhibit long-range order without translational symmetry. These structures challenge traditional paradigms of condensed matter physics and open avenues for discovering new critical behaviors. I present original contributions that combine analytical methods and numerical algorithms to study constrained systems on quasicrystalline tilings. These include the solution of NP-hard optimization problems like the Hamiltonian cycle on Ammann-Beenker tilings, an energy-based worm algorithm for simulating classical dimers on random graphs and modified Penrose tilings, and the analytical solution of the dimer partition function on the newly discovered aperiodic Spectre tiling. I further extend the study to quantum systems by analytically investigating the quantum dimer model (QDM) on the Spectre tiling and numerically analyzing it on the Ammann-Beenker tiling, revealing distinctive ground-state properties and quantum phase transitions that arise due to aperiodicity and local tiling constraints. This work builds interdisciplinary bridges between condensed matter physics, graph theory, and computational complexity. It also sheds light on the feasibility and limitations of transferring universal results from periodic systems to aperiodic ones.