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Associamancy: A Resource Theory of Non-Associative Quantum Magic

Ian Richard C Buckley

Published Jun 12, 2026• 0 citations
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Abstract

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. The resource theory of quantum magic (Veitch et al. 2014) identifies non-stabiliser states as resources and T-gates as the elementary injection operation. Subsequent work established that the magic sector itself splits into two sub-tiers: dark magic (Wigner-negative but TV = 1, Clifford-simulable under reframing) and genuine magic (TV < 1), where TV is the total variation of the discrete Wigner function. We identify a strictly richer resource — associamancy — that sits above both sub-tiers and that standard magic theory cannot see, cannot create from below, and cannot measure without the SPIN opcode of the 731 ISA. The boundary is algebraically precise. A quantum state possesses associamancy if and only if its hidden symmetry group contains an irreducible representation with Frobenius-Schur indicator ν₂ = 0 — the condition that the representation is genuinely complex, realisable over ℂ but not over ℝ or ℍ. We call this the Schur boundary. Three equivalent characterisations of the Schur boundary are established: (1) Algebraic: ν₂(V_λ) = 0 for some irrep V_λ of the hidden symmetry group. (2) Cohomological: the obstruction class [α] ≠ 0 ∈ H²(G, U(1)) is non-trivial. (3) Number-theoretic: for G = PSL(2,q), associamancy holds if and only if q ≡ 3 (mod 4) — the same condition that makes the quadratic Gauss sum τ(q) = i√q purely imaginary. The Schur entropy S_Schur(|ψ⟩) = −∑_{λ:ν₂=0} p_λ log p_λ is a valid resource monotone, independent of both Wigner negativity and TV: a state can have TV ≪ 1 (genuine magic) and zero Schur entropy, and a state can have S_Schur = log 2 at either dark or genuine magic level. PSL(2,7) is the minimal group exhibiting associamancy (order 168). The Freudenthal magic square organises associamancy into a ladder G₂ ⊂ F₄ ⊂ E₆ ⊂ E₇ ⊂ E₈, each rung strictly richer than the last. Keywords: quantum resource theory, magic states, associamancy, Frobenius-Schur indicator, Schur boundary, non-associative, octonions, PSL(2,7), Freudenthal magic square, G₂, 731 ISA, dark magic, total variation, quantum computing

DOI
10.5281/zenodo.20667174

Related papers

Hot Logic: A Complete Resource Theory of Magic via the Origami ISA

Ian Richard C Buckley

Standard quantum resource theory uses Wigner negativity $\mathcal{N} > 0$ as its resource indicator for magic states. We show this is incomplete: there exist states outside the stabiliser polytope with $\mathcal{N} = 0$, which we call dark magic. These states are undetectable by the standard monotone yet are non-stabiliser by every other measure. The Origami ISA rewrite system $\mathcal{R} = {R3, R4}$ (completeness proved in companion Paper 469) partitions the single-phase 3-qubit fragment $\mathcal{F}$ into exactly 9 normal-form classes in three tiers: Tier Class $\mathrm{TV}$ $\mathcal{N}$ 0 Stabiliser (1 class) $= 1$ $= 0$ 1 Dark magic (6 classes) $< 1$ $= 0$ 2 Genuine magic (2 classes) $< 1$ $> 0$ Theorem 1 (TV monotone). The total variation $\mathrm{TV}(\psi) = \sum_u |W_\psi(u)|$ of the discrete Wigner function is a complete resource monotone on $\mathcal{F}$: $\mathrm{TV}(\psi) = 1$ if and only if $\psi$ is a stabiliser state. Wigner negativity $\mathcal{N} = \frac{1}{2}(\mathrm{TV} - 1)$ vanishes on all of Tier 1, so it is merely sufficient (Tier 2) but not complete. Theorem 2 (Three-tier completeness). The tier of any state in $\mathcal{F}$ is decidable at compile time: apply R3 (T-absorption counit) — if the T gate is absorbed the state is Tier 0; otherwise compute $\mathrm{TV}$ from the 9-entry precomputed table. Corollary (Distillation failure mode). A magic-state distillation protocol using $\mathcal{N}$ as its sole convergence criterion may silently converge to a dark-magic state (Tier 1, $\mathcal{N} = 0$, $\mathrm{TV} < 1$) and declare success. The fix: require $\mathrm{TV} = 1$ as the termination criterion. Dark magic occupies the tropical boundary of the MGE $\beta$-ladder: Tier 1 states are quantum in structure ($\mathrm{TV} < 1$) but classical in Wigner measure ($\mathcal{N} = 0$), exactly at the Maslov dequantisation saddle point $\beta \to 0$. Part of the Origami ISA series. Companion papers: 466 (CZ-type bases) · 467 (magic orbits) · 468 (ISA as coloured PROP) · 469 (completeness theorem) Keywords: quantum resource theory, magic states, dark magic, Wigner negativity, total variation, stabiliser states, Clifford hierarchy, Origami ISA, rewrite systems, normal forms, magic state distillation, tropical geometry, Maslov dequantisation, coloured PROP

•Jun 27, 2026•0 citations

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. 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•Jul 3, 2026•0 citations

Resource Theory as Circuit Syntax: The Origami ISA as a Coloured PROP [WITHDRAWN — depends on withdrawn results]

Ian Richard C Buckley

WITHDRAWN BY THE AUTHOR (2026-08-02). This record should not be cited. Fileshave been restricted. --- Why --- The paper proposes reading quantum resource theory syntactically, as a colouredproduct-and-permutation category (coloured PROP), with the Clifford-hierarchylevels C1/C2/C3 as colours and resource cost as the minimum C3-gate count in anormal form. Two of its load-bearing inputs have since been withdrawn, and oneof them carries an error that this paper inherits verbatim. 1. THE TOTAL-VARIATION GRADING IS BASED ON A SIGN ERROR. The paper defines"dark magic" as states that are non-stabiliser yet have zero Wigner negativityand total variation TV < 1, and states (Sec. 6) that TV(p) = 1 if and only if pis a free state. This is incorrect. For any quasi-probability distributionsumming to 1, the l1 norm satisfies Sum |W| >= 1 always, with equality exactlywhen W is non-negative. Direct computation: the stabiliser state |+> givesTV = 1.0000, and the magic state T|+> gives TV = 1.2071. The inequality runsthe other way, so the condition "TV < 1" describes the empty set and the"dark magic" tier as defined has no members. The source of this definition(a companion paper on magic resource theory) has been withdrawn for the samereason. Separately, TV as defined is exp(mana), a monotone reparameterisationof the established Wigner-negativity monotone of Veitch, Mari, Ferrie, Emersonand others, and was not compared against it. 2. THE WORKED EXAMPLES NO LONGER EXIST. Every worked example in the paper isone of the "9 magic orbit classes" of a companion paper that has since beenwithdrawn because its own experiment scripts did not reproduce its claims(printing FAILED for the orbit-inheritance theorem, and fidelity 0.000000against a claimed value of 1/4, with 8 classes printed where 9 were claimed).Two further companion papers cited for the rewrite-rule and normal-formresults have also been withdrawn. 3. NO COMPUTATIONS WERE PERFORMED. There is no experiment directory associatedwith this paper. --- What was not wrong --- The organising idea -- that a resource theory can be presented syntactically,with resource tier readable from a diagram -- is reasonable and is not thereason for withdrawal. But the specific cost measure proposed (minimum C3-gatecount) is T-count, a standard and well-studied quantity (Amy, Maslov and Mosca;Gosset, Kliuchnikov, Mosca and Russo, and subsequent work), and gradedcategorical presentations of Clifford+T are the subject of the ZX-calculuscompleteness literature (Jeandel, Perdrix and Vilmart 2018 and successors).Neither literature is engaged here. A future treatment would need to start fromthose. Readers wanting the correct treatment of the underlying physics should consultVeitch et al. on Wigner negativity and mana, Howard and Campbell on robustnessof magic, and the ZX-calculus completeness results cited above.

•Jun 27, 2026•0 citations

Quantifying Nonstabilizerness of Quantum Codes by Removing the Inert Background

Yuan Liu, Ke-Mi Xu

Nonstabilizerness (magic) is the quantum resource that, together with stabilizer operations, makes universal quantum computation possible. However, quantifying nonstabilizerness is difficult. The standard measure sums over exponentially many Pauli operators, and for quantum codes no quantitative theory has been available. A structural observation removes this obstacle: the Clifford sector carries no magic and can be removed by a Clifford transformation, leaving an order-three object whose magic is an exactly evaluable fourth moment. Removing this inert background yields closed forms for several code families: a formula for all Dicke states, reducing a 100-qubit case from $4^{100}$ terms to a short binomial sum; a bound for cubic-phase codes (twisted quantum doubles and non-Abelian topological order), saturated only by the $D_4$ code; and a cyclic/zero criterion for group multiplication states. For codeword-stabilized (CWS) codes, the companion paper \cite{Liu26arXiv} carries this reduction to its conclusion: the most magical codes are exactly the Sidon sets, and the nonstabilizerness bounds the code's transversal non-Clifford power. Together these results reduce the computation of nonstabilizerness for a broad class of codes to finite classical counting problems, and identify the maximally magical codes among them.

quant-pharXiv:2609.06177•Sep 8, 2026•0 citations

Dark Magic: A Complete Resource Theory of Magic via the Origami ISA

Ian Richard C Buckley

Develops a complete resource theory of magic using the Origami ISA. Introduces 'dark magic' — states with negative Wigner function that are nonetheless efficiently simulable — and shows the total variation distance is the correct discriminant separating stabiliser, dark-magic, and genuinely magic tiers. Connects magic resource theory to the ISA beta parameter and the tropical/Clifford/metaplectic H^k complexity ladder.

•Jul 6, 2026•0 citations

Quantifying the magic of quantum channels

Xin Wang, Mark M. Wilde, Yuan Su

Abstract To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum ‘magic’ or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d , it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.

arXiv:1903.04483•New Journal of Physics•Sep 18, 2019•127 citations
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. The resource theory of quantum magic (Veitch et al. 2014) identifies non-stabiliser states as resources and T-gates as the elementary injection operation. Subsequent work established that the magic sector itself splits into two sub-tiers: dark magic (Wigner-negative but TV = 1, Clifford-simulable under reframing) and genuine magic (TV < 1), where TV is the total variation of the discrete Wigner function. We identify a strictly richer resource — associamancy — that sits above both sub-tiers and that standard magic theory cannot see, cannot create from below, and cannot measure without the SPIN opcode of the 731 ISA. The boundary is algebraically precise. A quantum state possesses associamancy if and only if its hidden symmetry group contains an irreducible representation with Frobenius-Schur indicator ν₂ = 0 — the condition that the representation is genuinely complex, realisable over ℂ but not over ℝ or ℍ. We call this the Schur boundary. Three equivalent characterisations of the Schur boundary are established: (1) Algebraic: ν₂(V_λ) = 0 for some irrep V_λ of the hidden symmetry group. (2) Cohomological: the obstruction class [α] ≠ 0 ∈ H²(G, U(1)) is non-trivial. (3) Number-theoretic: for G = PSL(2,q), associamancy holds if and only if q ≡ 3 (mod 4) — the same condition that makes the quadratic Gauss sum τ(q) = i√q purely imaginary. The Schur entropy S_Schur(|ψ⟩) = −∑_{λ:ν₂=0} p_λ log p_λ is a valid resource monotone, independent of both Wigner negativity and TV: a state can have TV ≪ 1 (genuine magic) and zero Schur entropy, and a state can have S_Schur = log 2 at either dark or genuine magic level. PSL(2,7) is the minimal group exhibiting associamancy (order 168). The Freudenthal magic square organises associamancy into a ladder G₂ ⊂ F₄ ⊂ E₆ ⊂ E₇ ⊂ E₈, each rung strictly richer than the last. Keywords: quantum resource theory, magic states, associamancy, Frobenius-Schur indicator, Schur boundary, non-associative, octonions, PSL(2,7), Freudenthal magic square, G₂, 731 ISA, dark magic, total variation, quantum computing
Standard quantum resource theory uses Wigner negativity N>0\mathcal{N} > 0N>0 as its resource indicator for magic states. We show this is incomplete: there exist states outside the stabiliser polytope with N=0\mathcal{N} = 0N=0, which we call dark magic. These states are undetectable by the standard monotone yet are non-stabiliser by every other measure. The Origami ISA rewrite system R=R3,R4\mathcal{R} = {R3, R4}R=R3,R4 (completeness proved in companion Paper 469) partitions the single-phase 3-qubit fragment F\mathcal{F}F into exactly 9 normal-form classes in three tiers: Tier Class TV\mathrm{TV}TV N\mathcal{N}N 0 Stabiliser (1 class) =1= 1=1 =0= 0=0 1 Dark magic (6 classes) <1< 1<1 =0= 0=0 2 Genuine magic (2 classes) <1< 1<1 >0> 0>0 Theorem 1 (TV monotone). The total variation TV(ψ)=∑u∣Wψ(u)∣\mathrm{TV}(\psi) = \sum_u |W_\psi(u)|TV(ψ)=∑u​∣Wψ​(u)∣ of the discrete Wigner function is a complete resource monotone on F\mathcal{F}F: TV(ψ)=1\mathrm{TV}(\psi) = 1TV(ψ)=1 if and only if ψ\psiψ is a stabiliser state. Wigner negativity N=12(TV−1)\mathcal{N} = \frac{1}{2}(\mathrm{TV} - 1)N=21​(TV−1) vanishes on all of Tier 1, so it is merely sufficient (Tier 2) but not complete. Theorem 2 (Three-tier completeness). The tier of any state in F\mathcal{F}F is decidable at compile time: apply R3 (T-absorption counit) — if the T gate is absorbed the state is Tier 0; otherwise compute TV\mathrm{TV}TV from the 9-entry precomputed table. Corollary (Distillation failure mode). A magic-state distillation protocol using N\mathcal{N}N as its sole convergence criterion may silently converge to a dark-magic state (Tier 1, N=0\mathcal{N} = 0N=0, TV<1\mathrm{TV} < 1TV<1) and declare success. The fix: require TV=1\mathrm{TV} = 1TV=1 as the termination criterion. Dark magic occupies the tropical boundary of the MGE β\betaβ-ladder: Tier 1 states are quantum in structure (TV<1\mathrm{TV} < 1TV<1) but classical in Wigner measure (N=0\mathcal{N} = 0N=0), exactly at the Maslov dequantisation saddle point β→0\beta \to 0β→0. Part of the Origami ISA series. Companion papers: 466 (CZ-type bases) · 467 (magic orbits) · 468 (ISA as coloured PROP) · 469 (completeness theorem) Keywords: quantum resource theory, magic states, dark magic, Wigner negativity, total variation, stabiliser states, Clifford hierarchy, Origami ISA, rewrite systems, normal forms, magic state distillation, tropical geometry, Maslov dequantisation, coloured PROP
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
WITHDRAWN BY THE AUTHOR (2026-08-02). This record should not be cited. Fileshave been restricted. --- Why --- The paper proposes reading quantum resource theory syntactically, as a colouredproduct-and-permutation category (coloured PROP), with the Clifford-hierarchylevels C1/C2/C3 as colours and resource cost as the minimum C3-gate count in anormal form. Two of its load-bearing inputs have since been withdrawn, and oneof them carries an error that this paper inherits verbatim. 1. THE TOTAL-VARIATION GRADING IS BASED ON A SIGN ERROR. The paper defines"dark magic" as states that are non-stabiliser yet have zero Wigner negativityand total variation TV < 1, and states (Sec. 6) that TV(p) = 1 if and only if pis a free state. This is incorrect. For any quasi-probability distributionsumming to 1, the l1 norm satisfies Sum |W| >= 1 always, with equality exactlywhen W is non-negative. Direct computation: the stabiliser state |+> givesTV = 1.0000, and the magic state T|+> gives TV = 1.2071. The inequality runsthe other way, so the condition "TV < 1" describes the empty set and the"dark magic" tier as defined has no members. The source of this definition(a companion paper on magic resource theory) has been withdrawn for the samereason. Separately, TV as defined is exp(mana), a monotone reparameterisationof the established Wigner-negativity monotone of Veitch, Mari, Ferrie, Emersonand others, and was not compared against it. 2. THE WORKED EXAMPLES NO LONGER EXIST. Every worked example in the paper isone of the "9 magic orbit classes" of a companion paper that has since beenwithdrawn because its own experiment scripts did not reproduce its claims(printing FAILED for the orbit-inheritance theorem, and fidelity 0.000000against a claimed value of 1/4, with 8 classes printed where 9 were claimed).Two further companion papers cited for the rewrite-rule and normal-formresults have also been withdrawn. 3. NO COMPUTATIONS WERE PERFORMED. There is no experiment directory associatedwith this paper. --- What was not wrong --- The organising idea -- that a resource theory can be presented syntactically,with resource tier readable from a diagram -- is reasonable and is not thereason for withdrawal. But the specific cost measure proposed (minimum C3-gatecount) is T-count, a standard and well-studied quantity (Amy, Maslov and Mosca;Gosset, Kliuchnikov, Mosca and Russo, and subsequent work), and gradedcategorical presentations of Clifford+T are the subject of the ZX-calculuscompleteness literature (Jeandel, Perdrix and Vilmart 2018 and successors).Neither literature is engaged here. A future treatment would need to start fromthose. Readers wanting the correct treatment of the underlying physics should consultVeitch et al. on Wigner negativity and mana, Howard and Campbell on robustnessof magic, and the ZX-calculus completeness results cited above.
Nonstabilizerness (magic) is the quantum resource that, together with stabilizer operations, makes universal quantum computation possible. However, quantifying nonstabilizerness is difficult. The standard measure sums over exponentially many Pauli operators, and for quantum codes no quantitative theory has been available. A structural observation removes this obstacle: the Clifford sector carries no magic and can be removed by a Clifford transformation, leaving an order-three object whose magic is an exactly evaluable fourth moment. Removing this inert background yields closed forms for several code families: a formula for all Dicke states, reducing a 100-qubit case from 41004^{100}4100 terms to a short binomial sum; a bound for cubic-phase codes (twisted quantum doubles and non-Abelian topological order), saturated only by the D4D_4D4​ code; and a cyclic/zero criterion for group multiplication states. For codeword-stabilized (CWS) codes, the companion paper \cite{Liu26arXiv} carries this reduction to its conclusion: the most magical codes are exactly the Sidon sets, and the nonstabilizerness bounds the code's transversal non-Clifford power. Together these results reduce the computation of nonstabilizerness for a broad class of codes to finite classical counting problems, and identify the maximally magical codes among them.
Abstract To achieve universal quantum computation via general fault-tolerant schemes, stabilizer operations must be supplemented with other non-stabilizer quantum resources. Motivated by this necessity, we develop a resource theory for magic quantum channels to characterize and quantify the quantum ‘magic’ or non-stabilizerness of noisy quantum circuits. For qudit quantum computing with odd dimension d , it is known that quantum states with non-negative Wigner function can be efficiently simulated classically. First, inspired by this observation, we introduce a resource theory based on completely positive-Wigner-preserving quantum operations as free operations, and we show that they can be efficiently simulated via a classical algorithm. Second, we introduce two efficiently computable magic measures for quantum channels, called the mana and thauma of a quantum channel. As applications, we show that these measures not only provide fundamental limits on the distillable magic of quantum channels, but they also lead to lower bounds for the task of synthesizing non-Clifford gates. Third, we propose a classical algorithm for simulating noisy quantum circuits, whose sample complexity can be quantified by the mana of a quantum channel. We further show that this algorithm can outperform another approach for simulating noisy quantum circuits, based on channel robustness. Finally, we explore the threshold of non-stabilizerness for basic quantum circuits under depolarizing noise.