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Fibrational Tensor Codes (FTCs): Holonomic Quantum Computation, $G_2$ Fiber Bundles, and the Cayley-Dickson Hardware Hierarchy

Ian Richard C Buckley

Published Apr 27, 2026• 0 citations
Publisher

Abstract

NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form. The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other. The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1*w2*w3*w4 and then checks w1*w2*w3*w4*w5 = +1, i.e. x*x = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true". What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing. Additional note. This record predates a change of vocabulary and, in places, of substance. Opcode names used here were subsequently revised (ORBIT to RESOLVE, LABEL to PROJECT, BIND to FUSE, MERGE to JOIN), so the terminology does not match the current reference. Readers wanting the current definitions should consult the maintained opcode reference rather than this record. This working paper introduces Fibrational Tensor Codes (FTCs), an exploratory quantum error correction paradigm that relocates the burden of error correction from macroscopic spatial topology — as in surface codes — to microscopic gauge topology. Rather than arranging qubits on 2D associative grids, the FTC architecture maps the Cayley-Dickson division algebra hierarchy directly onto hardware substrates: a Quaternionic routing layer (4 physical qubits, $[[4,2,2]]$, $PG(1,2)$) and an Octonionic computational layer (7 physical qubits, $PG(2,2)$, the Fano plane). The central conjecture proposes that the 7-qubit Hilbert space, isomorphic as a vector space to the Clifford algebra $\mathbb{C}\ell(0,7)$, can be physically confined to the 8-dimensional minimal left ideal identified by C. Furey in her construction of the complex octonions. This confinement — achieved by appropriate physical coupling rather than discrete stabilizer constraints — would yield a topologically protected 8-dimensional logical qudit (the Logical Quoct) in which local errors are converted to heralded erasures via an Associator Penalty, enabling active feed-forward correction at approximately twice the rate of blind Pauli-flip correction. The key architectural principle is that non-associativity is confined strictly to the continuous Berry phase of the encoding map within a $G_2$ fiber bundle; the logical gate group remains fully associative. The paper formally falsifies 7-qubit triorthogonality (Theorem 5.1), establishing that transversal CCZ requires scaling to the 15-qubit $PG(3,2)$ Reed-Muller geometry, and identifies six concrete open problems required to advance this framework from architectural proposal to engineering specification.

DOI
10.5281/zenodo.19821692

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Quantum Information Protocols Through the ISA Lens: Teleportation, Superdense Coding, and Bell Tests from H¹ to H³

Ian Richard C Buckley

PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. NOTICE (2026-08-04). This record uses a magic measure that has been shown to be incorrectly signed, and it should not be cited for quantitative magic claims. The "total variation" quantity TV = sum |W(u)| over the discrete Wigner function was used with the condition TV < 1 characterising non-stabiliser states. The inequality runs the other way. For any quasi-probability distribution summing to 1, the l1 norm satisfies sum |W| >= 1 always, with equality exactly when W is non-negative. Direct computation: the stabiliser state |+> gives TV = 1.0000 and the magic state T|+> gives TV = 1.2071. The condition "TV < 1" therefore describes the empty set, and any tier defined by it has no members. Separately, TV as defined is exp(mana) -- a monotone reparameterisation of the established Wigner-negativity monotone of Veitch, Mari, Ferrie and Emerson -- and was never compared against it. Readers should consult Veitch et al. on mana and Wigner negativity, Howard and Campbell on robustness of magic, Bravyi et al. on stabiliser extent, and Leone, Oliviero and Hamma on stabiliser Renyi entropy. NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form. The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other. The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1*w2*w3*w4 and then checks w1*w2*w3*w4*w5 = +1, i.e. x*x = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true". What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing. We recast the three foundational quantum information protocols — quantum teleportation, superdense coding, and the CHSH Bell test — in the language of the Origami Instruction Set Architecture (ISA), and then lift each to its TriQ (3-qubit GHZ) and 6-731 (broken-Fano-line) extensions. In the ISA reading, teleportation is SPLIT;SPLAT with a classical wire, superdense coding is FLIP;FLOP on a Bell pair encoding $\log_2|G|$ bits via the group $G = \mathbb{Z}_2^2$, and the CHSH violation is a hybrid H¹/H² computation: the Bell pair is a stabiliser (H¹) resource but the Tsirelson-optimal measurement applies a T-gate rotation (H², TWIST=1). We prove that the Tsirelson bound $\cos^2(\pi/8)$ equals the total variation of any dark-magic (Tier 1) state in the ISA magic-type classification — the Tsirelson bound is the Wigner mass of the dark-magic sector. Lifting to the TriQ register introduces the ORBIT opcode, raises superdense coding capacity to 3 bits per qubit ($\log_2|\mathbb{Z}_2^3|$), and replaces the CHSH game with the Fano Line Verification Game whose win conditions are the $\pm 1$ values of the $6j$ symbol of $\mathbb{Z}_2^3$. The 6-731 extension (one Fano line weakened to coupling $\eta$) makes teleportation directed, reduces superdense capacity to $3 - H(\eta)$ bits per qubit, and interpolates the Fano game win probability continuously between classical and quantum bounds. Remarkably, the same asymmetry parameter $\eta$ is independently measurable by three methods — thermodynamic Carnot efficiency (Paper 325), information-theoretic channel capacity, and game-theoretic win probability — whose agreement constitutes a Pentagon-identity self-verification test for the hardware. We close with the H$^k$ Bell tower: for each cohomological depth $k$ there is a Bell-type inequality whose maximum quantum violation requires an H$^k$ resource, providing a game-theoretic ladder that extends beyond CHSH through the Fano game (H²) to a conjectured G$_2$-correlation game (H³) accessible only to the 731-ISA SPIN opcode.Appendix B provides side-by-side quantum circuit and ISA spider diagrams for all three baseline protocols. Keywords quantum teleportation, superdense coding, CHSH inequality, Bell test, Tsirelson bound, Origami ISA, H^k cohomology, stabiliser states, magic states, Fano plane, GHZ state, TriQ register, 6-731 architecture, ORBIT opcode, TWIST opcode, topological heat engine, dark magic, Wigner mass, Bell hierarchy, Mermin inequality, Pentagon equation, quantum nonlocality, quantum channel capacity, G2 correlations

•Jul 3, 2026•0 citations

Réseaux de tenseurs holographiques d’Ising

Thomas Gobeil

La recherche de codes de correction d'erreurs est un point critique dans le développement d'un ordinateur quantique. Malheureusement, plusieurs des codes de correction trouvés jusqu'à maintenant n'ont qu'une valeur théorique, n'ayant pas de base dans un système physique qui peut être réalisé en laboratoire. Il est donc désirable de chercher un phénomène physique qui pourrait supporter un code de correction d'erreurs. Le principe holographique émergeant de certaines théories de gravité quantique, qui affirme qu'il existe une correspondance entre une théorie gravitationnelle dans un espace Anti-de Sitter de dimension $(d+1)$ et une théorie de champ conforme sur la frontière à $d$ dimensions de cet espace, est un candidat pour ce genre de phénomène. L'information locale dans la théorie gravitationnelle est représentée non-localement dans la théorie de champ, ouvrant la possibilité d'un code de correction d'erreurs. Un qubit logique est encodé dans la théorie gravitationnelle, les qubits physiques devenant des états dans la théorie de champ, soit une chaîne de spins dans la théorie d'Ising dans notre cas, qui est un système physiquement intéressant. Si une partie suffisamment petite des qubits physiques est effacée, le qubit logique peut quand même être reconstruit à partir des qubits physiques restants. Concrètement, cette correspondance est implémentée en utilisant des réseaux de tenseurs hyperboliques. Certains de ces réseaux permettent de représenter l'état appartenant à la théorie de champ conforme tout en possédant une structure interne qui détient les mêmes caractéristiques que l'espace AdS. Si une isométrie entre les tenseurs sur la frontière d'un tel réseau et le tenseur central du réseau existe, un code de correction d'erreurs devrait exister sur le réseau. La création de ces réseaux a été proposée par Jahn et al. en 2017. Ils ont démontré que l'utilisation de portes de parité, une classe de tenseurs introduite par Leslie Valiant en 2002 qui représente des états gaussiens, dans un réseau de tenseurs hyperbolique permet de retrouver un état propre d'un hamiltonien d'Ising à la frontière du réseau. Cependant, l'article qui décrit les réseaux hyperboliques de portes de parité n'offre pas de code explicite pour reproduire ces réseaux, et les explications contenues dans l'article sont parsemées de coquilles et omettent de l'information cruciale. Dans ce mémoire, ces problèmes sont résolus. Nous offrons d'abord une explication théorique plus accessible en introduisant les tenseurs et leurs réseaux, la correspondance AdS/CFT et le formalisme des portes de parité. Nous présentons également un algorithme permettant de recréer les réseaux discutés dans l'article de Jahn et al. Cette implémentation est générale, permettant de produire un réseau avec un dallage et une profondeur arbitraires tout en imposant les caractéristiques nécessaires à la correspondance AdS/CFT. L'algorithme inclut une fonction qui crée la liste d'adjacence du réseau de tenseurs ainsi que les fonctions requises pour manipuler les portes de parité et effectuer le passage de la liste d'adjacence au réseau de tenseurs contracté. De là, nous donnons également le code nécessaire pour calculer la matrice de covariance de l'état résultant, de laquelle tous les corrélateurs peuvent être extraits. Avec cet algorithme nous sommes capables de recréer les figures clés dans l'article de Jahn et al., confirmant la validité du code.

•Jan 1, 2020•0 citations

Holographic quantum error-correcting codes

Luis Carlos Mantilla Calderón

Los códigos correctores de errores cuánticos holográficos son una propuesta que intenta explicar, desde el punto de vista de la corrección de errores cuánticos, múltiples propiedades presentes en la correspondencia AdS / CFT. Esta tesis de pregrado proporcionará el material necesario para que el lector comprenda la motivación detrás de los códigos holográficos y estudiará la utilidad de esta familia de códigos. Nos centraremos en dos puntos particulares a lo largo del documento: (1) Discutir por qué este marco ayuda a explicar algunos resultados de correspondencia AdS / CFT, como la reconstrucción de AdS-Rindler y la fórmula de Ryu-Takayanagi. (2) Comprender si los códigos holográficos construidos (cuyo propósito principal es estudiar la holografía) son útiles, de alguna manera, para realizar la corrección de errores cuánticos. Para hacer esto, primero haremos una breve introducción a la mecánica cuántica para alguien sin tal experiencia. Luego, presentaremos los temas de corrección de errores cuánticos, los códigos estabilizadores y computación con tolerancia a fallas. Con esos temas en la mano, explicaremos la motivación detrás de los códigos holográficos y usaremos redes de tensores para construirlos. Posteriormente, calculamos las propiedades del código, como las tasas asintóticas para algunos de estos códigos y, finalmente, discutimos las ventajas y los desafíos de usar esta familia de códigos para realizar la corrección de errores en un computador cuántico.

•Jan 1, 2021•0 citations

[NOTICE — see description] The Architecture of Inevitability: The Freudenthal-Tits Magic Square, Exceptional Chains, and the Naturalness of Quantum Error Correction

Ian Richard C Buckley

CORRECTION NOTICE (2026-08). The Pachner/pentagon verification this framework relies on does not test what it appears to: the identical check passes on randomly assigned signs, with no octonions involved. A later direct test found the naive pentagon identity fails for octonionic labels in 1176 of 2401 cases. The structural framing survives — non-associativity makes rebracketing path-dependent by definition — but the associator must be supplied as explicit coherence data (Mac Lane; Kuperberg's spiders) rather than emerging from the verification claimed here. A revised version is in preparation. The design of fault-tolerant quantum computers (FTQC) is currently treated as an exercise in synthetic software engineering. Protocols such as the Steane $[[7,1,3]]$ code, the Reed-Muller $[[15,1,3]]$ code, and lattice surgery are widely viewed as human-invented algorithms imposed onto passive hardware substrates to artificially suppress noise. This paper fundamentally rejects that anthropocentric framing. We propose that these error-correcting codes are the computational shadows of innate geometric phase transitions within the Bruhat-Tits buildings of the exceptional Lie algebras. By mapping the Adelic Simplicial Architecture (ASA) portfolio to the Freudenthal-Tits Magic Square of normed division algebras ($\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$), we demonstrate that the ASA is complete by mathematical construction. We reframe fault tolerance not as an algorithmic imposition, but as the physical progression along the exceptional chain $G_2 \subset F_4 \subset E_6 \subset E_7$, where computational states thermodynamically retract into discrete, non-associative topological ground states.

•Apr 30, 2026•0 citations

Fault-tolerent Holonomic Quantum Computation Based on Stabilizer Codes

Yi-Cong Zheng, Todd A. Brun

Bulletin of the American Physical Society•Mar 19, 2013•0 citations

Non-Associative Quantum Error Correction (NA-QEC): The Exceptional Jordan-KL Condition and Rank-1 Idempotents in $G_2$ Topologies

Ian Richard C Buckley

CORRECTION NOTICE (2026-08). No error-correction simulation was performed for this work; the Knill–Laflamme conditions are asserted rather than verified. Treat the construction as a proposal. A revised version is in preparation. Standard quantum error correction (QEC) is founded on the Knill-Laflamme (KL) conditions, which implicitly assume that physical error operators compose associatively. Over the non-associative Octonions $\mathbb{O}$ and the exceptional Lie group $G_2$, this assumption fails: the elementary identity $(PE_a^\dagger)(E_b P) = P(E_a^\dagger E_b)P$ no longer holds, generating a chiral shear that breaks the Hermitian reality of observable eigenvalues. We resolve this obstruction by replacing ordinary matrix multiplication with the Jordan product $A\circ B = \frac{1}{2}(AB+BA)$, which restores Hermiticity by symmetrisation while retaining non-associativity. Working over the 27-dimensional Exceptional Jordan Algebra $\mathfrak{J}_3(\mathbb{O})$ (the Albert algebra), we prove the Exceptional Jordan-KL Condition: for the Furey projector $P = \mathrm{diag}(1,0,0) \in \mathfrak{J}_3(\mathbb{O})$, the Jordan U-operator satisfies $$\{P, E_a^\dagger \circ E_b, P\} = c_{ab} P$$ for every symmetrised error element $E_a^\dagger \circ E_b \in \mathfrak{J}_3(\mathbb{O})$, where $c_{ab} \in \mathbb{R}$ is a scalar determined by the normalised trace. The proof rests entirely on the standard Peirce decomposition of $\mathfrak{J}_3(\mathbb{O})$ and McCrimmon's classification of primitive idempotents; no novel algebraic machinery is required. We apply this result to the 731-QPU: the Steane $[[7,1,3]]$ code operates as the hardware-native QEC layer, and the Fano-plane incidence structure enforces distance $d=3$ geometrically via the Associator Penalty. NA-QEC monitors Jordan-algebraic coherence continuously rather than measuring discrete Pauli parities, trading measurement-induced decoherence for a geometric error-detection channel. We carefully delineate what is proved from what is conjectured: the claim that NA-QEC evades the Eastin-Knill theorem is stated as an open conjecture, motivated by the observation that the Eastin-Knill proof requires the logical subspace to carry a continuous Lie-group representation, a condition that is absent when the logical space is the 1-dimensional Peirce-1 subspace of $\mathfrak{J}_3(\mathbb{O})$.

•May 8, 2026•0 citations
Fibrational Tensor Codes (FTCs): Holonomic Quantum Computation, G2G_2G2​ Fiber Bundles, and the Cayley-Dickson Hardware Hierarchy
NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form. The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other. The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1w2w3w4 and then checks w1w2w3w4w5 = +1, i.e. xx = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true". What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing. Additional note. This record predates a change of vocabulary and, in places, of substance. Opcode names used here were subsequently revised (ORBIT to RESOLVE, LABEL to PROJECT, BIND to FUSE, MERGE to JOIN), so the terminology does not match the current reference. Readers wanting the current definitions should consult the maintained opcode reference rather than this record. This working paper introduces Fibrational Tensor Codes (FTCs), an exploratory quantum error correction paradigm that relocates the burden of error correction from macroscopic spatial topology — as in surface codes — to microscopic gauge topology. Rather than arranging qubits on 2D associative grids, the FTC architecture maps the Cayley-Dickson division algebra hierarchy directly onto hardware substrates: a Quaternionic routing layer (4 physical qubits, [[4,2,2]][[4,2,2]][[4,2,2]], PG(1,2)PG(1,2)PG(1,2)) and an Octonionic computational layer (7 physical qubits, PG(2,2)PG(2,2)PG(2,2), the Fano plane). The central conjecture proposes that the 7-qubit Hilbert space, isomorphic as a vector space to the Clifford algebra Cℓ(0,7)\mathbb{C}\ell(0,7)Cℓ(0,7), can be physically confined to the 8-dimensional minimal left ideal identified by C. Furey in her construction of the complex octonions. This confinement — achieved by appropriate physical coupling rather than discrete stabilizer constraints — would yield a topologically protected 8-dimensional logical qudit (the Logical Quoct) in which local errors are converted to heralded erasures via an Associator Penalty, enabling active feed-forward correction at approximately twice the rate of blind Pauli-flip correction. The key architectural principle is that non-associativity is confined strictly to the continuous Berry phase of the encoding map within a G2G_2G2​ fiber bundle; the logical gate group remains fully associative. The paper formally falsifies 7-qubit triorthogonality (Theorem 5.1), establishing that transversal CCZ requires scaling to the 15-qubit PG(3,2)PG(3,2)PG(3,2) Reed-Muller geometry, and identifies six concrete open problems required to advance this framework from architectural proposal to engineering specification.
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. NOTICE (2026-08-04). This record uses a magic measure that has been shown to be incorrectly signed, and it should not be cited for quantitative magic claims. The "total variation" quantity TV = sum |W(u)| over the discrete Wigner function was used with the condition TV < 1 characterising non-stabiliser states. The inequality runs the other way. For any quasi-probability distribution summing to 1, the l1 norm satisfies sum |W| >= 1 always, with equality exactly when W is non-negative. Direct computation: the stabiliser state |+> gives TV = 1.0000 and the magic state T|+> gives TV = 1.2071. The condition "TV < 1" therefore describes the empty set, and any tier defined by it has no members. Separately, TV as defined is exp(mana) -- a monotone reparameterisation of the established Wigner-negativity monotone of Veitch, Mari, Ferrie and Emerson -- and was never compared against it. Readers should consult Veitch et al. on mana and Wigner negativity, Howard and Campbell on robustness of magic, Bravyi et al. on stabiliser extent, and Leone, Oliviero and Hamma on stabiliser Renyi entropy. NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form. The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other. The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1*w2*w3*w4 and then checks w1*w2*w3*w4*w5 = +1, i.e. x*x = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true". What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing. We recast the three foundational quantum information protocols — quantum teleportation, superdense coding, and the CHSH Bell test — in the language of the Origami Instruction Set Architecture (ISA), and then lift each to its TriQ (3-qubit GHZ) and 6-731 (broken-Fano-line) extensions. In the ISA reading, teleportation is SPLIT;SPLAT with a classical wire, superdense coding is FLIP;FLOP on a Bell pair encoding log⁡2∣G∣\log_2|G|log2​∣G∣ bits via the group G=Z22G = \mathbb{Z}_2^2G=Z22​, and the CHSH violation is a hybrid H¹/H² computation: the Bell pair is a stabiliser (H¹) resource but the Tsirelson-optimal measurement applies a T-gate rotation (H², TWIST=1). We prove that the Tsirelson bound cos⁡2(π/8)\cos^2(\pi/8)cos2(π/8) equals the total variation of any dark-magic (Tier 1) state in the ISA magic-type classification — the Tsirelson bound is the Wigner mass of the dark-magic sector. Lifting to the TriQ register introduces the ORBIT opcode, raises superdense coding capacity to 3 bits per qubit (log⁡2∣Z23∣\log_2|\mathbb{Z}_2^3|log2​∣Z23​∣), and replaces the CHSH game with the Fano Line Verification Game whose win conditions are the ±1\pm 1±1 values of the 6j6j6j symbol of Z23\mathbb{Z}_2^3Z23​. The 6-731 extension (one Fano line weakened to coupling η\etaη) makes teleportation directed, reduces superdense capacity to 3−H(η)3 - H(\eta)3−H(η) bits per qubit, and interpolates the Fano game win probability continuously between classical and quantum bounds. Remarkably, the same asymmetry parameter η\etaη is independently measurable by three methods — thermodynamic Carnot efficiency (Paper 325), information-theoretic channel capacity, and game-theoretic win probability — whose agreement constitutes a Pentagon-identity self-verification test for the hardware. We close with the Hk^kk Bell tower: for each cohomological depth kkk there is a Bell-type inequality whose maximum quantum violation requires an Hk^kk resource, providing a game-theoretic ladder that extends beyond CHSH through the Fano game (H²) to a conjectured G2_22​-correlation game (H³) accessible only to the 731-ISA SPIN opcode.Appendix B provides side-by-side quantum circuit and ISA spider diagrams for all three baseline protocols. Keywords quantum teleportation, superdense coding, CHSH inequality, Bell test, Tsirelson bound, Origami ISA, H^k cohomology, stabiliser states, magic states, Fano plane, GHZ state, TriQ register, 6-731 architecture, ORBIT opcode, TWIST opcode, topological heat engine, dark magic, Wigner mass, Bell hierarchy, Mermin inequality, Pentagon equation, quantum nonlocality, quantum channel capacity, G2 correlations
La recherche de codes de correction d'erreurs est un point critique dans le développement d'un ordinateur quantique. Malheureusement, plusieurs des codes de correction trouvés jusqu'à maintenant n'ont qu'une valeur théorique, n'ayant pas de base dans un système physique qui peut être réalisé en laboratoire. Il est donc désirable de chercher un phénomène physique qui pourrait supporter un code de correction d'erreurs. Le principe holographique émergeant de certaines théories de gravité quantique, qui affirme qu'il existe une correspondance entre une théorie gravitationnelle dans un espace Anti-de Sitter de dimension (d+1)(d+1)(d+1) et une théorie de champ conforme sur la frontière à ddd dimensions de cet espace, est un candidat pour ce genre de phénomène. L'information locale dans la théorie gravitationnelle est représentée non-localement dans la théorie de champ, ouvrant la possibilité d'un code de correction d'erreurs. Un qubit logique est encodé dans la théorie gravitationnelle, les qubits physiques devenant des états dans la théorie de champ, soit une chaîne de spins dans la théorie d'Ising dans notre cas, qui est un système physiquement intéressant. Si une partie suffisamment petite des qubits physiques est effacée, le qubit logique peut quand même être reconstruit à partir des qubits physiques restants. Concrètement, cette correspondance est implémentée en utilisant des réseaux de tenseurs hyperboliques. Certains de ces réseaux permettent de représenter l'état appartenant à la théorie de champ conforme tout en possédant une structure interne qui détient les mêmes caractéristiques que l'espace AdS. Si une isométrie entre les tenseurs sur la frontière d'un tel réseau et le tenseur central du réseau existe, un code de correction d'erreurs devrait exister sur le réseau. La création de ces réseaux a été proposée par Jahn et al. en 2017. Ils ont démontré que l'utilisation de portes de parité, une classe de tenseurs introduite par Leslie Valiant en 2002 qui représente des états gaussiens, dans un réseau de tenseurs hyperbolique permet de retrouver un état propre d'un hamiltonien d'Ising à la frontière du réseau. Cependant, l'article qui décrit les réseaux hyperboliques de portes de parité n'offre pas de code explicite pour reproduire ces réseaux, et les explications contenues dans l'article sont parsemées de coquilles et omettent de l'information cruciale. Dans ce mémoire, ces problèmes sont résolus. Nous offrons d'abord une explication théorique plus accessible en introduisant les tenseurs et leurs réseaux, la correspondance AdS/CFT et le formalisme des portes de parité. Nous présentons également un algorithme permettant de recréer les réseaux discutés dans l'article de Jahn et al. Cette implémentation est générale, permettant de produire un réseau avec un dallage et une profondeur arbitraires tout en imposant les caractéristiques nécessaires à la correspondance AdS/CFT. L'algorithme inclut une fonction qui crée la liste d'adjacence du réseau de tenseurs ainsi que les fonctions requises pour manipuler les portes de parité et effectuer le passage de la liste d'adjacence au réseau de tenseurs contracté. De là, nous donnons également le code nécessaire pour calculer la matrice de covariance de l'état résultant, de laquelle tous les corrélateurs peuvent être extraits. Avec cet algorithme nous sommes capables de recréer les figures clés dans l'article de Jahn et al., confirmant la validité du code.
Los códigos correctores de errores cuánticos holográficos son una propuesta que intenta explicar, desde el punto de vista de la corrección de errores cuánticos, múltiples propiedades presentes en la correspondencia AdS / CFT. Esta tesis de pregrado proporcionará el material necesario para que el lector comprenda la motivación detrás de los códigos holográficos y estudiará la utilidad de esta familia de códigos. Nos centraremos en dos puntos particulares a lo largo del documento: (1) Discutir por qué este marco ayuda a explicar algunos resultados de correspondencia AdS / CFT, como la reconstrucción de AdS-Rindler y la fórmula de Ryu-Takayanagi. (2) Comprender si los códigos holográficos construidos (cuyo propósito principal es estudiar la holografía) son útiles, de alguna manera, para realizar la corrección de errores cuánticos. Para hacer esto, primero haremos una breve introducción a la mecánica cuántica para alguien sin tal experiencia. Luego, presentaremos los temas de corrección de errores cuánticos, los códigos estabilizadores y computación con tolerancia a fallas. Con esos temas en la mano, explicaremos la motivación detrás de los códigos holográficos y usaremos redes de tensores para construirlos. Posteriormente, calculamos las propiedades del código, como las tasas asintóticas para algunos de estos códigos y, finalmente, discutimos las ventajas y los desafíos de usar esta familia de códigos para realizar la corrección de errores en un computador cuántico.
CORRECTION NOTICE (2026-08). The Pachner/pentagon verification this framework relies on does not test what it appears to: the identical check passes on randomly assigned signs, with no octonions involved. A later direct test found the naive pentagon identity fails for octonionic labels in 1176 of 2401 cases. The structural framing survives — non-associativity makes rebracketing path-dependent by definition — but the associator must be supplied as explicit coherence data (Mac Lane; Kuperberg's spiders) rather than emerging from the verification claimed here. A revised version is in preparation. The design of fault-tolerant quantum computers (FTQC) is currently treated as an exercise in synthetic software engineering. Protocols such as the Steane [[7,1,3]][[7,1,3]][[7,1,3]] code, the Reed-Muller [[15,1,3]][[15,1,3]][[15,1,3]] code, and lattice surgery are widely viewed as human-invented algorithms imposed onto passive hardware substrates to artificially suppress noise. This paper fundamentally rejects that anthropocentric framing. We propose that these error-correcting codes are the computational shadows of innate geometric phase transitions within the Bruhat-Tits buildings of the exceptional Lie algebras. By mapping the Adelic Simplicial Architecture (ASA) portfolio to the Freudenthal-Tits Magic Square of normed division algebras (R,C,H,O\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}R,C,H,O), we demonstrate that the ASA is complete by mathematical construction. We reframe fault tolerance not as an algorithmic imposition, but as the physical progression along the exceptional chain G2⊂F4⊂E6⊂E7G_2 \subset F_4 \subset E_6 \subset E_7G2​⊂F4​⊂E6​⊂E7​, where computational states thermodynamically retract into discrete, non-associative topological ground states.
Non-Associative Quantum Error Correction (NA-QEC): The Exceptional Jordan-KL Condition and Rank-1 Idempotents in G2G_2G2​ Topologies
CORRECTION NOTICE (2026-08). No error-correction simulation was performed for this work; the Knill–Laflamme conditions are asserted rather than verified. Treat the construction as a proposal. A revised version is in preparation. Standard quantum error correction (QEC) is founded on the Knill-Laflamme (KL) conditions, which implicitly assume that physical error operators compose associatively. Over the non-associative Octonions O\mathbb{O}O and the exceptional Lie group G2G_2G2​, this assumption fails: the elementary identity (PEa†)(EbP)=P(Ea†Eb)P(PE_a^\dagger)(E_b P) = P(E_a^\dagger E_b)P(PEa†​)(Eb​P)=P(Ea†​Eb​)P no longer holds, generating a chiral shear that breaks the Hermitian reality of observable eigenvalues. We resolve this obstruction by replacing ordinary matrix multiplication with the Jordan product A∘B=12(AB+BA)A\circ B = \frac{1}{2}(AB+BA)A∘B=21​(AB+BA), which restores Hermiticity by symmetrisation while retaining non-associativity. Working over the 27-dimensional Exceptional Jordan Algebra J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) (the Albert algebra), we prove the Exceptional Jordan-KL Condition: for the Furey projector P=diag(1,0,0)∈J3(O)P = \mathrm{diag}(1,0,0) \in \mathfrak{J}_3(\mathbb{O})P=diag(1,0,0)∈J3​(O), the Jordan U-operator satisfies {P,Ea†∘Eb,P}=cabP\{P, E_a^\dagger \circ E_b, P\} = c_{ab} P{P,Ea†​∘Eb​,P}=cab​P for every symmetrised error element Ea†∘Eb∈J3(O)E_a^\dagger \circ E_b \in \mathfrak{J}_3(\mathbb{O})Ea†​∘Eb​∈J3​(O), where cab∈Rc_{ab} \in \mathbb{R}cab​∈R is a scalar determined by the normalised trace. The proof rests entirely on the standard Peirce decomposition of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) and McCrimmon's classification of primitive idempotents; no novel algebraic machinery is required. We apply this result to the 731-QPU: the Steane [[7,1,3]][[7,1,3]][[7,1,3]] code operates as the hardware-native QEC layer, and the Fano-plane incidence structure enforces distance d=3d=3d=3 geometrically via the Associator Penalty. NA-QEC monitors Jordan-algebraic coherence continuously rather than measuring discrete Pauli parities, trading measurement-induced decoherence for a geometric error-detection channel. We carefully delineate what is proved from what is conjectured: the claim that NA-QEC evades the Eastin-Knill theorem is stated as an open conjecture, motivated by the observation that the Eastin-Knill proof requires the logical subspace to carry a continuous Lie-group representation, a condition that is absent when the logical space is the 1-dimensional Peirce-1 subspace of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O).