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 Gottesman-Knill (GK) theorem establishes that Clifford circuits on stabiliser states are efficiently classically simulable. The simulability boundary is at zero magic: any state with positive Wigner negativity N(ρ) > 0 is beyond GK. This paper establishes that this boundary is not sharp enough. Two states can both lie beyond GK — both have N > 0 — yet differ in computational power in a precise, measurable way. The distinguishing invariant is the Fano orbit valence label {p_L}: the decomposition of the state's Wigner characteristic function over the seven orbit valences of the symplectic polar space W(5,2). The orbit valence provides a strictly finer simulability classification than total negativity N. The central new result is the magic valence mismatch theorem: injecting a magic state of the wrong orbit valence into a circuit that expects a different valence yields a winning probability of 0.472 in the Fano Line Verification Game — strictly below the 0.500 classical bound. Wrong-valence magic is not merely suboptimal; it is worse than no magic at all for the intended computation. Critically, this failure mode is invisible to syndrome-only decoders: CS₀₁, CS₀₂, and CS₁₂ states all produce identical stabiliser syndromes (+1)^7 but distinct orbit valence labels, detectable only by the ORBIT opcode. A valence-aware magic state factory using the ORBIT opcode accepts states by orbit valence, achieving distillation overhead O(1/p) per accepted state rather than the O(1/p²) of standard T-gate distillation — a ~1000× overhead reduction at physical error rate p = 10⁻³ for CS-family computations. The island graph picture of quantum circuit computation is introduced: entangled registers form islands connected by magic-carrying channels, and the computational power of the circuit is determined not by total magic N but by the magic current profile {I_L}. The ORBIT opcode serves as an ammeter for magic current: 7 Pauli measurements identify which orbit valences are flowing at each node. Keywords Gottesman-Knill Theorem, Classical Simulation, Magic State, Wigner Negativity, Fano Orbit, Magic Valence, Orbit Valence Label, W(5,2), Symplectic Polar Space, ORBIT Opcode, Magic State Distillation, CS Gate, Controlled-S Gate, Valence Mismatch, Island Graph, Magic Current, Fault-Tolerant Quantum Computation, Stabiliser Syndrome, Coherent Error Detection, Origami ISA, Magic-ISA, TriQ, SQU, Fano Plane, PG(2,2), Quantum Resource Theory, Simulability Boundary
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 Gottesman-Knill (GK) theorem establishes that Clifford circuits on stabiliser states are efficiently classically simulable. The simulability boundary is at zero magic: any state with positive Wigner negativity N(ρ) > 0 is beyond GK. This paper establishes that this boundary is not sharp enough. Two states can both lie beyond GK — both have N > 0 — yet differ in computational power in a precise, measurable way. The distinguishing invariant is the Fano orbit valence label {p_L}: the decomposition of the state's Wigner characteristic function over the seven orbit valences of the symplectic polar space W(5,2). The orbit valence provides a strictly finer simulability classification than total negativity N. The central new result is the magic valence mismatch theorem: injecting a magic state of the wrong orbit valence into a circuit that expects a different valence yields a winning probability of 0.472 in the Fano Line Verification Game — strictly below the 0.500 classical bound. Wrong-valence magic is not merely suboptimal; it is worse than no magic at all for the intended computation. Critically, this failure mode is invisible to syndrome-only decoders: CS₀₁, CS₀₂, and CS₁₂ states all produce identical stabiliser syndromes (+1)^7 but distinct orbit valence labels, detectable only by the ORBIT opcode. A valence-aware magic state factory using the ORBIT opcode accepts states by orbit valence, achieving distillation overhead O(1/p) per accepted state rather than the O(1/p²) of standard T-gate distillation — a ~1000× overhead reduction at physical error rate p = 10⁻³ for CS-family computations. The island graph picture of quantum circuit computation is introduced: entangled registers form islands connected by magic-carrying channels, and the computational power of the circuit is determined not by total magic N but by the magic current profile {I_L}. The ORBIT opcode serves as an ammeter for magic current: 7 Pauli measurements identify which orbit valences are flowing at each node. Keywords Gottesman-Knill Theorem, Classical Simulation, Magic State, Wigner Negativity, Fano Orbit, Magic Valence, Orbit Valence Label, W(5,2), Symplectic Polar Space, ORBIT Opcode, Magic State Distillation, CS Gate, Controlled-S Gate, Valence Mismatch, Island Graph, Magic Current, Fault-Tolerant Quantum Computation, Stabiliser Syndrome, Coherent Error Detection, Origami ISA, Magic-ISA, TriQ, SQU, Fano Plane, PG(2,2), Quantum Resource Theory, Simulability Boundary
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. This paper develops the island graph picture of quantum circuit computation. At any time step, a quantum circuit partitions into entanglement islands — maximal sets of mutually entangled qubits forming totally isotropic subspaces of the symplectic polar space W(2k-1,2). Each island carries a Fano orbit valence label {p_L} — its magic valence. Magic flows between islands via non-Clifford gates, typed by the XOR-Fano composition rule. The central claim: entanglement determines the topology of the island graph (which qubits are correlated), while magic determines the computational content (what the computation can achieve beyond classical simulation). These are independent. Within the island graph, magic current obeys a diode law: primary-orbit magic (L ∈ {1,2,3}) flows naturally toward secondary-orbit states (L ∈ {4,5,6}) via the 6-7 broken-Fano topology, with a measurable asymmetry ratio D decreasing monotonically from 6/7 (stabiliser state) to approximately 0.650 (triple-CS state). Wrong-valence magic — primary-orbit magic used in a secondary-orbit context — scores below the classical bound (0.472 < 0.500), making magic valence mismatch an experimentally detectable computational error. The ORBIT opcode serves as a magic ammeter: seven Pauli measurements on a TriQ identify the orbit valence label at any node, providing real-time monitoring of magic flow throughout the circuit at zero additional syndrome cost. The paper connects the island graph picture to the biological FMO complex: the bacteriochlorophyll network is a natural 6-7 topology whose directed energy transport is magic flow at biological temperatures, with the same diode asymmetry ratio measured experimentally. Keywords Magic Flow, Island Graph, Magic Current, Orbit Valence, Fano Orbit, Magic Valence, ORBIT Opcode, Magic Ammeter, 6-7 Topology, Magic Diode, XOR-Fano Rule, Entanglement, Quantum Circuit, W(5,2), Symplectic Polar Space, Wigner Negativity, TriQ, SQU, Origami ISA, FMO Complex, Bacteriochlorophyll, Biological Quantum Coherence, Fault-Tolerant Quantum Computation, Valence Mismatch, CS Gate, Non-Clifford Gates, Adelic Simplicial Architecture
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. Nonlocal games provide an operational framework for distinguishing quantum from classical resources: quantum players sharing entanglement can win certain games with higher probability than any classical strategy. This paper introduces magic-valence nonlocal games — a family of games whose winning probability is determined not by entanglement alone but by the Fano orbit valence of the quantum state shared between players. The central result is that the seven Fano orbit valences {L=0,...,6} of the 3-qubit symplectic polar space W(5,2) correspond to seven distinct winning-probability tiers in the Fano Line Verification Game. Stabiliser states (L=0) win at the classical bound 1/7. Each non-classical orbit valence L≥1 achieves a strictly higher winning probability ω_L ∈ (1/7, 1], with the maximum achieved by states with full secondary-orbit support {4,5,6}. The magic valence mismatch theorem is proved: a player who injects magic of the wrong orbit valence — primary-orbit magic in a game requiring secondary-orbit magic — scores 0.472, strictly below the 0.500 classical bound. Wrong-valence magic is operationally worse than no magic for the intended game. This gives a sharp operational definition of magic valence: it is the resource that determines game-winning ability, not just total Wigner negativity. Five Clifford-equivalence classes of CS-family magic states are identified and their game distinguishability characterised. The maximum game distinguishability of 7/8 is proved: no single game can distinguish all seven orbit valences simultaneously, but 7 out of 8 can be separated by the Fano Line Verification Game family. The paper establishes the connection between magic-valence nonlocal games and the ORBIT opcode: the 7 stabiliser measurements of ORBIT are precisely the measurements needed to determine which game tier a state belongs to, at zero additional circuit cost beyond syndrome extraction. Keywords Nonlocal Games, Magic Valence, Fano Orbit, Orbit Valence, Quantum Advantage, Fano Line Verification Game, W(5,2), Symplectic Polar Space, Wigner Negativity, Magic State, Valence Mismatch, ORBIT Opcode, CS Gate, Clifford Equivalence, Game Distinguishability, Stabiliser States, Quantum Resource Theory, Bell Inequality, Fano Plane, PG(2,2), Origami ISA, Magic-ISA, TriQ, Classical Simulation Bound
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 standard resource theory of magic treats magic as a scalar quantity — the total Wigner negativity of a quantum state. This paper shows that for 3-qubit systems, magic has additional structure: a valence, encoded in a 7-component orbit label indexed by the seven lines of the Fano plane PG(2,2). The 63-point 3-qubit symplectic phase space W(5,2) decomposes into the Fano plane (7 classical points) plus 7 orbits of 8 points each under the GHZ stabiliser group; each orbit corresponds to a distinct Fano line and carries a distinct magic valence. Just as chemical valence determines what bonds an atom can form, the magic valence of a quantum state determines what gates it can natively implement and what other magic states it can combine with. The central result is the CS Magic Composition Theorem: for any subset of controlled-S (CS) gates applied to the all-plus state, the negative-chi support is determined exactly by a symmetric-difference rule on the Fano geometry. Single CS gates produce primary-orbit magic; pairs produce secondary-orbit magic via the XOR-Fano rule; all three CS gates together produce magic exclusively in the three secondary orbits. The valence composition rule is bitwise XOR in Z₂³ — the Clebsch-Gordan rule of the representation ring R(Z₂³). The orbit label is stated in terms of the Pauli characteristic function χ rather than the Gross-Wigner function — a necessity, not a limitation, as a theorem of Raussendorf, Okay, Zurel and Feldmann establishes cohomological obstructions to Clifford-covariant Wigner functions for qubits. The characteristic function χ is the canonical magic indicator forced by these obstructions. Operational consequences: the orbit valence label distinguishes coherent gate-type errors (wrong CS gate applied to the wrong qubit pair) that are invisible to total negativity. The 7 stabiliser measurements of the Fano self-test simultaneously certify the GHZ state and read the magic valence label at zero additional circuit cost. Note: Earlier records (v1.0) used the term "magic type." Version 1.1 adopts the preferred term "magic valence" throughout, and adds a footnote connecting the concept to Queen's A Kind of Magic (1986). The mathematics is identical.
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 Eastin-Knill theorem states that no quantum error correcting code can have a universal set of transversal gates. This is widely interpreted as requiring magic-state distillation (MSD) as an unavoidable overhead for fault-tolerant non-Clifford computation. This paper demonstrates three distinct routes to CS-family non-Clifford gates on the SQU architecture that circumvent the practical impact of Eastin-Knill without violating it. Route 1 — Gauge-orbit CS transversality: The [[7,1,3]] Steane code has 6 gauge degrees of freedom corresponding bijectively to the 6 non-classical Fano orbit valences. Preparing the gauge subsystem in orbit valence L ∈ {1,2,3} via the DISTIL opcode enables transversal application of CS_{ij} — the qubit pair associated with orbit L — across all 7 physical qubit triplets simultaneously. This implements a logical CS gate fault-tolerantly at 7 physical qubits, versus 15 for the [[15,1,3]] Reed-Muller approach, with zero magic-state distillation and zero code switching. Route 2 — CONVERT as free valence switching: The CONVERT opcode is a free Clifford gate that switches orbit valence within the primary family {1,2,3} or secondary family {4,5,6}. Any CS-type gate is reachable from any other via one SWAP gate, at zero distillation cost. The gauge-orbit correspondence makes CS-family universality a matter of gauge selection, not resource consumption. Route 3 — Syndrome-blind valence verification: The CS₀₁, CS₀₂, and CS₁₂ states are indistinguishable by stabiliser syndrome (all produce syndrome (+1)⁷) but are perfectly distinguished by the ORBIT opcode. This enables fault-tolerant gate verification at zero additional syndrome cost — the ORBIT measurement is performed on the same 7 qubits already measured for error correction. Together these routes reduce the physical qubit overhead for CS-family computation from 15 to 7, eliminate MSD entirely for CS gates, and provide built-in coherent error detection. The Eastin-Knill theorem is respected: the CS gate is not transversal in the code sense. What changes is the architecture — the gauge structure of the Steane code is the resource, not ancilla magic states. Keywords Eastin-Knill Theorem, Transversal Gates, Magic State Distillation, Fault-Tolerant Quantum Computation, CS Gate, Controlled-S Gate, Steane Code, [[7,1,3]] Code, Gauge Selection, Fano Orbit Valence, Orbit Valence, ORBIT Opcode, DISTIL Opcode, CONVERT Opcode, Magic-ISA, SQU Architecture, TriQ, Quantum Error Correction, Syndrome Blindness, Non-Clifford Gates, Gauge Subsystem, W(5,2), Fano Plane, PG(2,2), Origami ISA, Zero Overhead, Coherent Error Detection
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. Einstein's "spooky action at a distance" conflates two distinct phenomena: statistical non-locality (correlations that violate Bell inequalities) and computational non-classicality (correlations that cannot be efficiently simulated classically). These are not the same thing. This paper shows that quantum non-classicality has two independent components: entanglement topology (which parties are correlated, measured by concurrence or entanglement entropy) and magic valence (what kind of non-classical correlations they share, measured by the Fano orbit label {p_L}). The Bell/CHSH framework captures the first component but not the second. The Fano Line Verification Game captures both. Crucially: a maximally entangled Bell pair |Φ⁺⟩ = (|00⟩ + |11⟩)/√2 is a stabiliser state — it has orbit label L=0 and zero magic. It violates Bell inequalities but is efficiently classically simulable by the Gottesman-Knill theorem. Computationally meaningful "spookiness" — the kind that enables quantum advantage — requires magic (non-zero orbit valence), not just entanglement. The Spacelike Associator Paradox, Hardy's Paradox, and the Fano Monogamy Paradox are revisited through this two-component lens. Each is sharpened: the ORBIT opcode provides a concrete experimental test distinguishing entanglement non-locality from magic non-locality in a single 7-measurement protocol. Einstein's objection to spooky action is reinterpreted: he objected to entanglement non-locality alone — the second component, magic non-locality, was not accessible to him. Keywords Spooky Action, Quantum Non-Locality, Entanglement, Magic Valence, Fano Orbit, Bell Inequality, CHSH Inequality, Gottesman-Knill Theorem, Magic State, Wigner Negativity, ORBIT Opcode, Fano Line Verification Game, W(5,2), Quantum Advantage, Stabiliser State, Einstein, Hardy's Paradox, Entanglement Topology, Magic Non-Locality, Origami ISA, TriQ, PG(2,2), Quantum Resource Theory, Classical Simulation, Non-Classicality
Non-Clifford gates are essential for avoiding classical simulability under the Gottesman-Knill theorem. An error-corrected circuit with non-Clifford gates will therefore be an important element for realizing useful fault-tolerant quantum algorithms. However, accurately understanding the fault-tolerant performance of such circuits faces challenges due to the classical hardness of simulating these circuits. We avoid simulating logical measurements and develop exact and efficient classical methods for benchmarking circuits composed of X, CNOT, and diagonal gates in the Clifford hierarchy, including state and gate teleportation gadgets with feedforward. For circuits with third-level diagonal gates, our benchmarking algorithm efficiently samples syndrome statistics and estimates the output logical fidelity and expectation values of certain Clifford observables, including all Pauli observables. The evaluation of syndrome expectation values also extends to circuits with fourth-level diagonal gates. Our algorithm scales polynomially in the number of physical qubits n and circuit depth T, independent of the number of logical qubits. The benchmarking applications include IQP sampling and magic state preparation/cultivation. We demonstrate the practical utility of our simulation algorithms by using them to optimize high-rate magic state distillation factories based on three copies of a [[27,3,3]] tricycle code.