The Projective Hierarchy of Computation: Classical, Clifford, and Magic as Shadows of the Meld

Ian Richard C Buckley

Published Jul 6, 2026• 0 citations

Abstract

Establishes the H^k complexity ladder as a projective hierarchy: H0 (classical/tropical) and H1 (Clifford/stabiliser) are projective shadows of H2 (magic/metaplectic). The Meld — the H2 completion of the ISA — projects onto Origami (H0+H1 shadow) and Forge (H1 shadow) in the same way that a 3D object projects onto 2D shadows. Derives the Palmer Invariant Space-time conjecture as the fixed-point condition of Penrose objective reduction within this framework.

DOI: 10.5281/zenodo.21219706

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Information Geometry as the H^k Ladder: Fisher Metrics, Alpha-Connections, and Quantum Holonomy in the Origami ISA

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Maps the full programme of information geometry onto the H^k ISA ladder. H0 corresponds to flat exponential families and tropical likelihood (A(theta) as tropical potential); H1 to Amari's alpha-connection and TWIST monodromy; H2 to Uhlmann holonomy and BIND. The EM algorithm is a SPLIT/SPLAT cycle; the beta* snap is the maximum Fisher information threshold. Provides a unified ISA syntax for classical and quantum information geometry.

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The Three Forms of the Maslov-Gibbs Einsum: Tropical, Clifford, and Metaplectic

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Shows that the Maslov-Gibbs Einsum (MGE) takes three canonical forms corresponding to the H0/H1/H2 complexity tiers: tropical (min-plus at beta = infinity), Clifford/geometric-Kitaev (at beta = beta*), and metaplectic/Maslov-index (at beta = 0 or imaginary). Establishes the Maslov duality theorem connecting H0 and H2 forms via Legendre transform. Introduces the Penrose empire analogy for H2 non-locality and identifies an H3 frontier.

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Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic

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The Projective Hierarchy of Computation: Classical, Clifford, and Magic as Shadows of the Meld

Ian Richard C Buckley

Published Jul 6, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.21219706

Related papers

The Meld ISA: Complex-MGE, Quantum Algorithm Discovery, and the T-Gate as Octonion Obstruction

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. We complete the TRS trilogy — Origami ( $\beta\to\infty$ , tropical, classical), Forge ( $0<\beta<\infty$ , real, statistical), Meld ( $\beta=it$ , complex, quantum) — by showing that the Meld ISA is the quantum circuit model expressed in the TRS semiring language. Three results follow. Result 1 (ISA completeness). The Clifford group equals the Meld ISA without the BIND opcode. Universal quantum computation equals Meld plus BIND. The T-gate $T = \mathrm{diag}(1,, e^{i\pi/4})$ is the BIND opcode at $j=\tfrac{1}{2}$ — the Fano associator obstruction of Paper 407, applied to qubit representations. The magic-typing classification of Papers 365/366 maps exactly onto the 731-ISA opcode hierarchy: Clifford gates occupy Fano-line orbits (SPLIT/FLIP/TWIST at Clifford angles); the T-gate occupies the non-collinear Fano triple (BIND). Result 2 (Quantum algorithm factory). Every classical algorithm of the form $\arg\max_{g\in G} f(g)$ over an abelian group $G$ has a $\mathbb{C}$ -MGE quantum counterpart $\mathbb{C}\text{-MGE}(f) = \mathrm{SPLIT}_G \cdot \mathrm{FLIP}(f) \cdot \mathrm{SPLAT}$ , giving exponential speedup via quantum Fourier sampling whenever $G$ is abelian and $f$ has a hidden subgroup concentrating Fourier mass. Shor's algorithm decomposes into three ISA layers: Layer 1 (Origami — classical modular exponentiation), Layer 2 (Meld — QFT as SPLIT-TWIST-FLIP cascade), Layer 3 (Origami — measurement as tropical projection). "Frozen-Shor" (Origami ISA, $\beta\to\infty$ ) recovers classical exhaustive search at cost $O(N)$ . "Warm-Shor" (Forge ISA, real $\beta$ ) gives Grover-like $O(\sqrt{N})$ thermal concentration. Full Shor (Meld, $\beta=it$ ) achieves $O(\log N)$ via destructive interference — a resource unavailable in either real ISA. New algorithm candidates follow by identifying the symmetry group of the target problem: equivariant ML, quantum persistent homology (Lloyd-Garnerone-Zanardi 2016), and Boson Sampling (Aaronson-Arkhipov 2011) all fit the five-step factory template. Result 3 (ML-KEM structural no-go). ML-KEM (Module-LWE) is immune because LWE noise exponentially suppresses every Fourier component: $\mathbb{E}{\mathbf{e}\sim D\sigma}[e^{i\langle\xi,\mathbf{e}\rangle}] = e^{-|\xi|^2\sigma^2/2}$ . No Fourier peak exists; SPLAT cannot extract a period. The deeper reason is that worst-case LWE reduces to the dihedral hidden subgroup problem (Regev 2002) — a non-abelian problem for which no efficient Meld program is known. Breaking ML-KEM would require a BIND opcode capable of distinguishing dihedral cosets, a capability not established for the current 731-Meld. The $H^k$ complexity ladder (Paper 420) is realised by the trilogy: $H^0$ (tree) = Origami, $H^1$ (loop) = Forge/BPP, $H^2$ (surface) = Meld/BQP. The T-gate is the $H^2$ generator that lifts computation off the stabiliser submanifold into the curved interior of the Fisher information metric. Series: Paper 258 (Origami ISA) → Paper 419 (Forge ISA) → Paper 443 ( $\oplus_\beta$ semiring, six pairs) → Paper 453 (multi-sector Forge) → Paper 454 (this paper — Meld ISA, trilogy complete). Related identifiers (Cites) Paper 258 Origami ISA: doi:10.5281/zenodo.19916429 Paper 419 Forge ISA: doi:10.5281/zenodo.20694527 Paper 443 ⊕_β semiring: doi:10.5281/zenodo.20752384 Paper 407 Associamancy: doi:10.5281/zenodo.20702223 Paper 365 Fano Magic Factory: doi:10.5281/zenodo.20541583 Paper 366 CS Magic Composition: doi:10.5281/zenodo.20541665 Paper 445 Kitaev anyons: doi:10.5281/zenodo.20752352 Paper 453 Multi-sector Forge: doi:10.5281/zenodo.20773423 Paper 420 H^k complexity ladder: doi:10.5281/zenodo.20773526PENDING 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. We complete the TRS trilogy — Origami ($\beta\to\infty$, tropical, classical), Forge ($0<\beta<\infty$, real, statistical), Meld ($\beta=it$, complex, quantum) — by showing that the Meld ISA is the quantum circuit model expressed in the TRS semiring language. Three results follow. Result 1 (ISA completeness). The Clifford group equals the Meld ISA without the BIND opcode. Universal quantum computation equals Meld plus BIND. The T-gate $T = \mathrm{diag}(1,, e^{i\pi/4})$ is the BIND opcode at $j=\tfrac{1}{2}$ — the Fano associator obstruction of Paper 407, applied to qubit representations. The magic-typing classification of Papers 365/366 maps exactly onto the 731-ISA opcode hierarchy: Clifford gates occupy Fano-line orbits (SPLIT/FLIP/TWIST at Clifford angles); the T-gate occupies the non-collinear Fano triple (BIND). Result 2 (Quantum algorithm factory). Every classical algorithm of the form $\arg\max_{g\in G} f(g)$ over an abelian group $G$ has a $\mathbb{C}$-MGE quantum counterpart $\mathbb{C}\text{-MGE}(f) = \mathrm{SPLIT}_G \cdot \mathrm{FLIP}(f) \cdot \mathrm{SPLAT}$, giving exponential speedup via quantum Fourier sampling whenever $G$ is abelian and $f$ has a hidden subgroup concentrating Fourier mass. Shor's algorithm decomposes into three ISA layers: Layer 1 (Origami — classical modular exponentiation), Layer 2 (Meld — QFT as SPLIT-TWIST-FLIP cascade), Layer 3 (Origami — measurement as tropical projection). "Frozen-Shor" (Origami ISA, $\beta\to\infty$) recovers classical exhaustive search at cost $O(N)$. "Warm-Shor" (Forge ISA, real $\beta$) gives Grover-like $O(\sqrt{N})$ thermal concentration. Full Shor (Meld, $\beta=it$) achieves $O(\log N)$ via destructive interference — a resource unavailable in either real ISA. New algorithm candidates follow by identifying the symmetry group of the target problem: equivariant ML, quantum persistent homology (Lloyd-Garnerone-Zanardi 2016), and Boson Sampling (Aaronson-Arkhipov 2011) all fit the five-step factory template. Result 3 (ML-KEM structural no-go). ML-KEM (Module-LWE) is immune because LWE noise exponentially suppresses every Fourier component: $\mathbb{E}{\mathbf{e}\sim D\sigma}[e^{i\langle\xi,\mathbf{e}\rangle}] = e^{-|\xi|^2\sigma^2/2}$. No Fourier peak exists; SPLAT cannot extract a period. The deeper reason is that worst-case LWE reduces to the dihedral hidden subgroup problem (Regev 2002) — a non-abelian problem for which no efficient Meld program is known. Breaking ML-KEM would require a BIND opcode capable of distinguishing dihedral cosets, a capability not established for the current 731-Meld. The $H^k$ complexity ladder (Paper 420) is realised by the trilogy: $H^0$ (tree) = Origami, $H^1$ (loop) = Forge/BPP, $H^2$ (surface) = Meld/BQP. 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Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic

Ryohei Kobayashi, Guanyu Zhu, Po-Shen Hsin

A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for $n$ -dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of $n$ -dimensional Clifford hierarchy stabilizer codes. These codes correspond to the $(n+1)$ D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted $\mathbb{Z}_2^3$ gauge theory (equivalent to $\mathbb{D}_4$ topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in $O(d)$ rounds via code switching. In 3D, we construct a transversal logical $\sqrt{\text{T}}$ gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted $\mathbb{Z}_2^4$ gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the $(n+1)$ -th level of Clifford hierarchy in $n$ spatial dimension.A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for $n$-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of $n$-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the $(n+1)$D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted $\mathbb{Z}_2^3$ gauge theory (equivalent to $\mathbb{D}_4$ topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in $O(d)$ rounds via code switching. In 3D, we construct a transversal logical $\sqrt{\text{T}}$ gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted $\mathbb{Z}_2^4$ gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the $(n+1)$-th level of Clifford hierarchy in $n$ spatial dimension.

quant-pharXiv:2511.02900Physical Review LettersNov 6, 20251 citation

The Three Forms of the Maslov-Gibbs Einsum: H⁰ Tropical, H¹ Clifford, H² Metaplectic

Ian Richard C Buckley

The Maslov-Gibbs Einsum (MGE) is not a single algorithm but a family of tensor contraction schemes indexed by the cohomological level $H^k$ of the physical system. This paper identifies three distinct forms: $H^0$ MGE (classical/tropical): contraction over the min-plus tropical semiring, corresponding to Maslov dequantisation ( $\hbar \to 0$ ). Finds the dominant classical path; computational class P. $H^1$ MGE (Clifford/stabiliser): contraction over the stabiliser calculus of the Clifford group $\mathsf{C}_2$ . Captures quantum interference within the stabiliser sector. Classically simulable in polynomial time by the Gottesman-Knill theorem, but exponentially more expressive than $H^0$ : captures non-local correlations, entanglement, and quantum error correction. $H^2$ MGE (metaplectic/magic): contraction with Maslov index phase corrections. Each gate composition $g \circ h$ acquires a $\mathbb{Z}_2$ phase factor $(-1)^{\tau(L, gL, ghL)}$ from the Maslov index of three Lagrangian subspaces. The TWIST opcode in the Origami ISA counts how many times this correction fires; TWIST $> 0$ is necessary and sufficient for $H^2$ MGE. Computational class BQP-hard. The central observation is that Viktor Maslov named both ends of this ladder: Maslov dequantisation ( $H^0$ , 1990s) and the Maslov index ( $H^2$ , 1965) are not coincidentally related — they are the two limits of the same quantisation programme, with the Clifford/Berry-phase level $H^1$ interpolating between them. The name "Maslov-Gibbs Einsum" was more prophetic than intended: Maslov's index governs the $H^2$ correction as well as the $H^0$ tropical limit. The Origami ISA compiler selects the MGE form automatically: TWIST $= 0$ implies $H^1$ MGE suffices; TWIST $> 0$ requires $H^2$ MGE. This makes the Schur multiplier $\mathbb{Z}_2$ a compile-time certificate for quantum advantage. The paper also introduces the Penrose empire problem as a non-quantum illustration of why $H^0$ local consistency checking cannot detect $H^1$ and $H^2$ obstructions, and discusses the $H^3$ frontier (Dijkgraaf-Witten state sums, Fibonacci anyon braiding) as the next level beyond current qubit hardware. Keywords Maslov-Gibbs Einsum, tropical semiring, Maslov dequantisation, Maslov index, Clifford group, stabiliser calculus, metaplectic group, cohomological filtration, $H^k$ tower, TWIST opcode, Origami ISA, Gottesman-Knill theorem, quantum advantage, BQP, magic states, tensor contraction, Penrose tiling, Dijkgraaf-Witten, topological quantum computationThe Maslov-Gibbs Einsum (MGE) is not a single algorithm but a family of tensor contraction schemes indexed by the cohomological level $H^k$ of the physical system. This paper identifies three distinct forms: $H^0$ MGE (classical/tropical): contraction over the min-plus tropical semiring, corresponding to Maslov dequantisation ($\hbar \to 0$). Finds the dominant classical path; computational class P. $H^1$ MGE (Clifford/stabiliser): contraction over the stabiliser calculus of the Clifford group $\mathsf{C}_2$. Captures quantum interference within the stabiliser sector. Classically simulable in polynomial time by the Gottesman-Knill theorem, but exponentially more expressive than $H^0$: captures non-local correlations, entanglement, and quantum error correction. $H^2$ MGE (metaplectic/magic): contraction with Maslov index phase corrections. Each gate composition $g \circ h$ acquires a $\mathbb{Z}_2$ phase factor $(-1)^{\tau(L, gL, ghL)}$ from the Maslov index of three Lagrangian subspaces. The TWIST opcode in the Origami ISA counts how many times this correction fires; TWIST $> 0$ is necessary and sufficient for $H^2$ MGE. Computational class BQP-hard. The central observation is that Viktor Maslov named both ends of this ladder: Maslov dequantisation ($H^0$, 1990s) and the Maslov index ($H^2$, 1965) are not coincidentally related — they are the two limits of the same quantisation programme, with the Clifford/Berry-phase level $H^1$ interpolating between them. The name "Maslov-Gibbs Einsum" was more prophetic than intended: Maslov's index governs the $H^2$ correction as well as the $H^0$ tropical limit. The Origami ISA compiler selects the MGE form automatically: TWIST $= 0$ implies $H^1$ MGE suffices; TWIST $> 0$ requires $H^2$ MGE. This makes the Schur multiplier $\mathbb{Z}_2$ a compile-time certificate for quantum advantage. The paper also introduces the Penrose empire problem as a non-quantum illustration of why $H^0$ local consistency checking cannot detect $H^1$ and $H^2$ obstructions, and discusses the $H^3$ frontier (Dijkgraaf-Witten state sums, Fibonacci anyon braiding) as the next level beyond current qubit hardware. Keywords Maslov-Gibbs Einsum, tropical semiring, Maslov dequantisation, Maslov index, Clifford group, stabiliser calculus, metaplectic group, cohomological filtration, $H^k$ tower, TWIST opcode, Origami ISA, Gottesman-Knill theorem, quantum advantage, BQP, magic states, tensor contraction, Penrose tiling, Dijkgraaf-Witten, topological quantum computation

Jul 3, 20260 citations

The Generalized Semi-Clifford Conjecture Holds at Level 4

Maxwell Marcus, Sathyawageeswar Subramanian, Marcel Dall'Agnol

The Clifford hierarchy $\mathsf{C}_1 \subset \mathsf{C}_2 \subset \cdots$ was introduced by Gottesman and Chuang (arXiv:quant-ph/9908010) to characterize gates that admit fault-tolerant implementation by gate teleportation. Yet, despite its rich mathematical structure and the attention it has received in recent years, little is known about $\mathsf{C}_k$ for $k > 3$ . Most progress has focused on identifying structural properties of restrictions of the hierarchy, such as diagonal gates and gates on systems of small dimension $d$ or with few qudits. The generalized semi-Clifford conjecture, proposed by Zeng et al. (arXiv:0712.2084), states that every gate in $\cup_k \mathsf{C}_k$ is, up to multiplication by Cliffords, the product of a permutation and a diagonal matrix. Beigi and Shor proved the case $d=2$ , $k=3$ (arXiv:0810.5108) and Pllaha et al. found an alternative proof by exploiting fixed points of the conjugation map induced by (a Clifford correction of) $U \in \mathsf{C}_3$ on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group $Γ_1(U)$ generated by $U \mathsf{P} U^\dagger$ and beyond, we prove the conjecture for $k \leq 4$ and any prime dimension $d$ . Our proof centers on conjugation groups $Γ_1(U), Γ_2(U), \ldots$ of $U \in \mathsf{C}_k$ , which we expect to be a useful tool in the study of the Clifford hierarchy more generally. We also show a natural sufficient condition on such groups for gates in higher levels to be generalized semi-Clifford.The Clifford hierarchy $\mathsf{C}_1 \subset \mathsf{C}_2 \subset \cdots$ was introduced by Gottesman and Chuang (arXiv:quant-ph/9908010) to characterize gates that admit fault-tolerant implementation by gate teleportation. Yet, despite its rich mathematical structure and the attention it has received in recent years, little is known about $\mathsf{C}_k$ for $k > 3$. Most progress has focused on identifying structural properties of restrictions of the hierarchy, such as diagonal gates and gates on systems of small dimension $d$ or with few qudits. The generalized semi-Clifford conjecture, proposed by Zeng et al. (arXiv:0712.2084), states that every gate in $\cup_k \mathsf{C}_k$ is, up to multiplication by Cliffords, the product of a permutation and a diagonal matrix. Beigi and Shor proved the case $d=2$, $k=3$ (arXiv:0810.5108) and Pllaha et al. found an alternative proof by exploiting fixed points of the conjugation map induced by (a Clifford correction of) $U \in \mathsf{C}_3$ on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group $Γ_1(U)$ generated by $U \mathsf{P} U^\dagger$ and beyond, we prove the conjecture for $k \leq 4$ and any prime dimension $d$. Our proof centers on conjugation groups $Γ_1(U), Γ_2(U), \ldots$ of $U \in \mathsf{C}_k$, which we expect to be a useful tool in the study of the Clifford hierarchy more generally. We also show a natural sufficient condition on such groups for gates in higher levels to be generalized semi-Clifford.

quant-pharXiv:2609.38751Oct 1, 20260 citations

\beta\to\infty

, tropical, classical), Forge (

0<\beta<\infty

, real, statistical), Meld (

\beta=it

, complex, quantum) — by showing that the Meld ISA is the quantum circuit model expressed in the TRS semiring language. Three results follow. Result 1 (ISA completeness). The Clifford group equals the Meld ISA without the BIND opcode. Universal quantum computation equals Meld plus BIND. The T-gate

T = \mathrm{diag}(1,, e^{i\pi/4})

is the BIND opcode at

j=\tfrac{1}{2}

— the Fano associator obstruction of Paper 407, applied to qubit representations. The magic-typing classification of Papers 365/366 maps exactly onto the 731-ISA opcode hierarchy: Clifford gates occupy Fano-line orbits (SPLIT/FLIP/TWIST at Clifford angles); the T-gate occupies the non-collinear Fano triple (BIND). Result 2 (Quantum algorithm factory). Every classical algorithm of the form

\arg\max_{g\in G} f(g)

over an abelian group

G

has a

\mathbb{C}

-MGE quantum counterpart

\mathbb{C}\text{-MGE}(f) = \mathrm{SPLIT}_G \cdot \mathrm{FLIP}(f) \cdot \mathrm{SPLAT}

, giving exponential speedup via quantum Fourier sampling whenever

G

is abelian and

f

has a hidden subgroup concentrating Fourier mass. Shor's algorithm decomposes into three ISA layers: Layer 1 (Origami — classical modular exponentiation), Layer 2 (Meld — QFT as SPLIT-TWIST-FLIP cascade), Layer 3 (Origami — measurement as tropical projection). "Frozen-Shor" (Origami ISA,

\beta\to\infty

) recovers classical exhaustive search at cost

O(N)

. "Warm-Shor" (Forge ISA, real

\beta

) gives Grover-like

O(\sqrt{N})

thermal concentration. Full Shor (Meld,

\beta=it

) achieves

O(\log N)

via destructive interference — a resource unavailable in either real ISA. New algorithm candidates follow by identifying the symmetry group of the target problem: equivariant ML, quantum persistent homology (Lloyd-Garnerone-Zanardi 2016), and Boson Sampling (Aaronson-Arkhipov 2011) all fit the five-step factory template. Result 3 (ML-KEM structural no-go). ML-KEM (Module-LWE) is immune because LWE noise exponentially suppresses every Fourier component:

\mathbb{E}{\mathbf{e}\sim D\sigma}[e^{i\langle\xi,\mathbf{e}\rangle}] = e^{-|\xi|^2\sigma^2/2}

. No Fourier peak exists; SPLAT cannot extract a period. The deeper reason is that worst-case LWE reduces to the dihedral hidden subgroup problem (Regev 2002) — a non-abelian problem for which no efficient Meld program is known. Breaking ML-KEM would require a BIND opcode capable of distinguishing dihedral cosets, a capability not established for the current 731-Meld. The

H^k

complexity ladder (Paper 420) is realised by the trilogy:

H^0

(tree) = Origami,

H^1

(loop) = Forge/BPP,

H^2

(surface) = Meld/BQP. The T-gate is the

H^2

generator that lifts computation off the stabiliser submanifold into the curved interior of the Fisher information metric. Series: Paper 258 (Origami ISA) → Paper 419 (Forge ISA) → Paper 443 (

\oplus_\beta

semiring, six pairs) → Paper 453 (multi-sector Forge) → Paper 454 (this paper — Meld ISA, trilogy complete). Related identifiers (Cites) Paper 258 Origami ISA: doi:10.5281/zenodo.19916429 Paper 419 Forge ISA: doi:10.5281/zenodo.20694527 Paper 443 ⊕_β semiring: doi:10.5281/zenodo.20752384 Paper 407 Associamancy: doi:10.5281/zenodo.20702223 Paper 365 Fano Magic Factory: doi:10.5281/zenodo.20541583 Paper 366 CS Magic Composition: doi:10.5281/zenodo.20541665 Paper 445 Kitaev anyons: doi:10.5281/zenodo.20752352 Paper 453 Multi-sector Forge: doi:10.5281/zenodo.20773423 Paper 420 H^k complexity ladder: doi:10.5281/zenodo.20773526

A fundamental problem in fault-tolerant quantum computation is the tradeoff between universality and dimensionality, exemplified by the the Bravyi-König bound for

n

-dimensional topological stabilizer codes. In this work, we extend topological Pauli stabilizer codes to a broad class of

n

-dimensional Clifford hierarchy stabilizer codes. These codes correspond to the

(n+1)

D Dijkgraaf-Witten gauge theories with non-Abelian topological order. We construct transversal non-Clifford gates through automorphism symmetries represented by cup products. In 2D, we obtain the first transversal non-Clifford logical gates including T and CS for Clifford stabilizer codes, using the automorphism of the twisted

\mathbb{Z}_2^3

gauge theory (equivalent to

\mathbb{D}_4

topological order). We also combine it with the just-in-time decoder to fault-tolerantly prepare the logical T magic state in

O(d)

rounds via code switching. In 3D, we construct a transversal logical

\sqrt{\text{T}}

gate in a non-Clifford stabilizer code at the third level of the Clifford hierarchy, located on a tetrahedron corresponding to a twisted

\mathbb{Z}_2^4

gauge theory. Our constructions surpass the Bravyi-König bound by achieving the logical gates in the

(n+1)

-th level of Clifford hierarchy in

n

spatial dimension.

The Maslov-Gibbs Einsum (MGE) is not a single algorithm but a family of tensor contraction schemes indexed by the cohomological level

H^k

of the physical system. This paper identifies three distinct forms:

H^0

MGE (classical/tropical): contraction over the min-plus tropical semiring, corresponding to Maslov dequantisation (

\hbar \to 0

). Finds the dominant classical path; computational class P.

H^1

MGE (Clifford/stabiliser): contraction over the stabiliser calculus of the Clifford group

\mathsf{C}_2

. Captures quantum interference within the stabiliser sector. Classically simulable in polynomial time by the Gottesman-Knill theorem, but exponentially more expressive than

H^0

: captures non-local correlations, entanglement, and quantum error correction.

H^2

MGE (metaplectic/magic): contraction with Maslov index phase corrections. Each gate composition

g \circ h

acquires a

\mathbb{Z}_2

phase factor

(-1)^{\tau(L, gL, ghL)}

from the Maslov index of three Lagrangian subspaces. The TWIST opcode in the Origami ISA counts how many times this correction fires; TWIST

> 0

is necessary and sufficient for

H^2

MGE. Computational class BQP-hard. The central observation is that Viktor Maslov named both ends of this ladder: Maslov dequantisation (

H^0

, 1990s) and the Maslov index (

H^2

, 1965) are not coincidentally related — they are the two limits of the same quantisation programme, with the Clifford/Berry-phase level

H^1

interpolating between them. The name "Maslov-Gibbs Einsum" was more prophetic than intended: Maslov's index governs the

H^2

correction as well as the

H^0

tropical limit. The Origami ISA compiler selects the MGE form automatically: TWIST

= 0

implies

H^1

MGE suffices; TWIST

> 0

requires

H^2

MGE. This makes the Schur multiplier

\mathbb{Z}_2

a compile-time certificate for quantum advantage. The paper also introduces the Penrose empire problem as a non-quantum illustration of why

H^0

local consistency checking cannot detect

H^1

and

H^2

obstructions, and discusses the

H^3

frontier (Dijkgraaf-Witten state sums, Fibonacci anyon braiding) as the next level beyond current qubit hardware. Keywords Maslov-Gibbs Einsum, tropical semiring, Maslov dequantisation, Maslov index, Clifford group, stabiliser calculus, metaplectic group, cohomological filtration,

H^k

tower, TWIST opcode, Origami ISA, Gottesman-Knill theorem, quantum advantage, BQP, magic states, tensor contraction, Penrose tiling, Dijkgraaf-Witten, topological quantum computation

The Clifford hierarchy

\mathsf{C}_1 \subset \mathsf{C}_2 \subset \cdots

was introduced by Gottesman and Chuang (arXiv:quant-ph/9908010) to characterize gates that admit fault-tolerant implementation by gate teleportation. Yet, despite its rich mathematical structure and the attention it has received in recent years, little is known about

\mathsf{C}_k

for

k > 3

. Most progress has focused on identifying structural properties of restrictions of the hierarchy, such as diagonal gates and gates on systems of small dimension

d

or with few qudits. The generalized semi-Clifford conjecture, proposed by Zeng et al. (arXiv:0712.2084), states that every gate in

\cup_k \mathsf{C}_k

is, up to multiplication by Cliffords, the product of a permutation and a diagonal matrix. Beigi and Shor proved the case

d=2

k=3

(arXiv:0810.5108) and Pllaha et al. found an alternative proof by exploiting fixed points of the conjugation map induced by (a Clifford correction of)

U \in \mathsf{C}_3

on the span of maximal stabilizer subgroups (arXiv:2006.14040). By extending their fixed-point arguments to the group

Γ_1(U)

generated by

U \mathsf{P} U^\dagger

and beyond, we prove the conjecture for

k \leq 4

and any prime dimension

d

. Our proof centers on conjugation groups

Γ_1(U), Γ_2(U), \ldots

U \in \mathsf{C}_k

, which we expect to be a useful tool in the study of the Clifford hierarchy more generally. We also show a natural sufficient condition on such groups for gates in higher levels to be generalized semi-Clifford.