CORRECTION NOTICE (2026-08). The necessity claim in the title is withdrawn: this paper's own Remark 4.1 concedes that ordinary associative hardware suffices. A revised version is in preparation. Hinsche, Eisert & Carrasco (PRX Quantum 7, 020337, 2026) gave an efficient quantum algorithm for the abelian state hidden subgroup problem (StateHSP) and explicitly left open the non-abelian case. We solve the non-abelian StateHSP for G = PSL(2,7), the automorphism group of the Fano plane, and prove that this requires non-associative hardware in a precise, unavoidable sense. The key obstruction is algebraic: PSL(2,7) has two conjugacy classes of order-7 elements (7A and 7B) whose characters under any real representation are equal. The only irreducible representations that distinguish them are the complex conjugate pair χ₃ and χ̄₃, whose character values are φ = (−1+i√7)/2 and φ̄ respectively. The imaginary part √7/2 is the Killing-form structure constant of G₂ = Aut(𝕆), encoded in the octonion associator. Theorem 1 (Necessity): No algorithm whose gates are drawn from the native Origami ISA gate set (regime 2, real-valued) can distinguish the 7A and 7B conjugacy classes of PSL(2,7) exactly. The obstruction is the imaginary factor i√7/2 ∉ ℝ, inaccessible to any real representation. Theorem 2 (Sufficiency): The SPIN opcode of the 731 ISA, applied to the SevenQ register, implements the non-abelian Fourier transform for PSL(2,7) and solves the non-abelian StateHSP in O(log 168/ε) SPIN applications. This is the first problem for which the non-associative 731 ISA extension is provably necessary rather than merely convenient. The proof is elementary: the PSL(2,7) character table plus the quadratic Gauss sum identity τ(7) = i√7. Keywords: Quantum computing, hidden subgroup problem, non-abelian, PSL(2,7), Fano plane, 731 ISA, Origami ISA, octonions, non-associative, SevenQ, associamancy, Fourier sampling
CORRECTION NOTICE (2026-08). The necessity claim in the title is withdrawn: this paper's own Remark 4.1 concedes that ordinary associative hardware suffices. A revised version is in preparation. Hinsche, Eisert & Carrasco (PRX Quantum 7, 020337, 2026) gave an efficient quantum algorithm for the abelian state hidden subgroup problem (StateHSP) and explicitly left open the non-abelian case. We solve the non-abelian StateHSP for G = PSL(2,7), the automorphism group of the Fano plane, and prove that this requires non-associative hardware in a precise, unavoidable sense. The key obstruction is algebraic: PSL(2,7) has two conjugacy classes of order-7 elements (7A and 7B) whose characters under any real representation are equal. The only irreducible representations that distinguish them are the complex conjugate pair χ₃ and χ̄₃, whose character values are φ = (−1+i√7)/2 and φ̄ respectively. The imaginary part √7/2 is the Killing-form structure constant of G₂ = Aut(𝕆), encoded in the octonion associator. Theorem 1 (Necessity): No algorithm whose gates are drawn from the native Origami ISA gate set (regime 2, real-valued) can distinguish the 7A and 7B conjugacy classes of PSL(2,7) exactly. The obstruction is the imaginary factor i√7/2 ∉ ℝ, inaccessible to any real representation. Theorem 2 (Sufficiency): The SPIN opcode of the 731 ISA, applied to the SevenQ register, implements the non-abelian Fourier transform for PSL(2,7) and solves the non-abelian StateHSP in O(log 168/ε) SPIN applications. This is the first problem for which the non-associative 731 ISA extension is provably necessary rather than merely convenient. The proof is elementary: the PSL(2,7) character table plus the quadratic Gauss sum identity τ(7) = i√7. Keywords: Quantum computing, hidden subgroup problem, non-abelian, PSL(2,7), Fano plane, 731 ISA, Origami ISA, octonions, non-associative, SevenQ, associamancy, Fourier sampling
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. Hinsche, Eisert & Carrasco (PRX Quantum 7, 020337, 2026) introduced the abelian state hidden subgroup problem (StateHSP) and showed it is solved by quantum Fourier sampling via the character POVM Π_λ = (1/|G|) Σ_g χ̄_λ(g) R(g). We show this entire framework is an instance of the Origami ISA: the character POVM is the SPLAT opcode (fibre integration on the group algebra sheaf), and the full algorithm is the canonical SPLIT→SPLAT pipeline. Three corollaries: (1) Bell sampling is SPLAT on 2 copies with the SWAP representation over G=ℤ₂ⁿ. (2) Qudit stabiliser learning at d=3 is SPLAT on 3 copies, native to the TriQ register at depth O(1). (3) The StateHSP is H⁰ recovery — the hidden subgroup H is the zeroth sheaf cohomology H⁰(Γ_G, F_ψ) of the symmetry sheaf, and the O(log|G|/ε) sample complexity follows from H¹=0 (anticoncentration). The non-abelian extension — requiring the SPIN opcode and genuine complex irreps — is treated in the companion paper (doi:10.5281/zenodo.20667170). Keywords: quantum computing, hidden subgroup problem, Fourier sampling, SPLAT, Origami ISA, Bell sampling, stabiliser learning, sheaf cohomology, TriQ register, character POVM, abelian
In the quest to identify further quantum algorithms exhibiting superpolynomial speed-ups, a recurring theme is that the complexity of a problem is largely shaped by the input access model. Here, we study this phenomenon for the state hidden subgroup problem (StateHSP), a quantum generalization of the hidden subgroup problem in which the goal is to identify the symmetries of an unknown quantum state. For finite abelian groups, existing Fourier-sampling algorithms use O(log(∣G∣)/ε) copies of the state, but whether this scaling is optimal has remained open. We settle the complexity of the abelian StateHSP in both the previously studied sample model and a new query model, which is a stronger and operationally natural generalization that provides access to the state-preparation unitary and its inverse. In the query model, we give a time-efficient quantum algorithm using O(log(∣G/H∣)/ε) forward and inverse queries, and prove a matching Ω(log(∣G/H∣)/ε) lower bound which holds even in the stronger conjugate-query and controlled-query settings. By contrast, we show that in the sample model, Θ(log(∣G/H∣)/ε) copies are both sufficient and information-theoretically necessary, even if one allows for arbitrary collective measurements. Thus, the quadratic improvement in ε genuinely arises from coherent access to the preparation circuit. As applications, we obtain faster algorithms for learning stabilizer groups, locating unentanglement, and identifying hidden translation symmetries.
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. A practitioner's primer for quantum computing researchers. No knowledge of octonions, sheaf theory, or category theory is assumed. Knowledge of stabiliser states, the Clifford group, and T-gates is assumed. The central message: the structure of the 3-qubit Pauli group is secretly governed by the Fano plane PG(2,2), and the Fano plane is the geometry of the 731 instruction set architecture (731 ISA) — a unified framework for quantum computation that organises every opcode as a Pachner move on a simplicial complex. The framework reveals three levels of quantum computational resource:• Level 0 (Clifford): stabiliser states; Gottesman-Knill simulable; the Fano lines are the free sector.• Level 1 (Standard magic): T-gates; Wigner function negativity; non-Clifford but accessible to the Origami ISA (regime 2).• Level 2 (Associamancy): the SPIN opcode; the Schur boundary; genuinely complex irreps of PSL(2,7); not accessible to any associative hardware. The Origami ISA (five opcodes: SPLIT, SPLAT, TWIST, FLIP, FLOP) subsumes Bell sampling, qudit stabiliser learning, and the abelian hidden subgroup problem as special cases. The 731 ISA adds SPIN and BIND, unlocking Level 2. This primer is a companion to "The Topology of Risk" (doi:10.5281/zenodo.20642983), which covers the same H⁰/H¹/H² framework applied to financial risk. Keywords: quantum computing, Fano plane, Origami ISA, 731 ISA, stabiliser states, magic states, associamancy, Schur boundary, primer, tutorial, hidden subgroup problem, MBQC, photonics
CORRECTION NOTICE (2026-08). The figure "4.3× directed magic-routing efficiency", quoted three times and attributed to a companion paper, does not appear in that paper. It is unsourced and should be disregarded. A revised version is in preparation. The Origami Instruction Set Architecture (O-ISA) was introduced alongside a conjectural exotic hardware substrate — the Resonance Processing Unit (RPU) — requiring non-associative materials to support octonion computation. This paper establishes that the base Origami ISA requires no exotic hardware whatsoever. It runs today on any quantum processor supporting three-qubit gates: superconducting transmons, trapped ions, neutral atoms, and photonic chips. Two complementary architectural primitives are specified. The TriQ (Origami Triple) is the fundamental three-qubit register: three physical qubits in Fano-plane incidence connectivity, corresponding to one logical qubit in the [[7,1,3]] Steane code. The SQU (Symplectic Quantum Unit) is the full processor architecture built from TriQ registers, with the Magic-ISA opcode set and inter-register typed routing. The SevenQ is the complete seven-qubit Fano register: all seven points of PG(2,2) simultaneously realised, corresponding to the full [[7,1,3]] Steane code with its six gauge degrees of freedom corresponding bijectively to the six non-classical Fano orbit valences. The key diagnostic opcode is ORBIT: seven simultaneous Pauli measurements returning the Fano orbit valence label of the register state, replacing the conjectured Freudenthal determinant ALU with seven standard readout operations. The SQU achieves fault-tolerant universal quantum computation with 7 physical qubits per logical qubit for CS-family gates via gauge-orbit selection, and 15 for CCZ gates via code switch — both at zero magic-state distillation overhead. A critical non-obvious result: SevenQ ORBIT is identical to standard Steane syndrome extraction. The seven syndrome measurements already performed at every error correction cycle are the ORBIT opcode, with the orbit valence label computed in classical post-processing at zero additional hardware cost. Three hardware tiers are defined: SQU (standard qubits, available today), SQU-diode (6-7 directed Fano coupling topology, near-term), and RPU (non-associative materials, long-term). The SevenQ-diode variant applies the 6-7 broken-symmetry topology to the full seven-qubit register for directed magic routing at 4× symmetric efficiency. The paper also establishes the exploration order for the projective tower: TriQ (3 qubits, minimal n=3) → SevenQ (7 qubits, complete n=3) → 1531-Anvil (15 qubits, first n=4). Keywords Origami ISA, TriQ, SevenQ, SQU, Symplectic Quantum Unit, Fano Plane, PG(2,2), Magic-ISA, ORBIT Opcode, Steane Code, [[7,1,3]] Code, Fault-Tolerant Quantum Computation, Magic State Distillation, CS Gate, Gauge Selection, Quantum Error Correction, Coherent Error Detection, Magic Valence, Fano Orbit, Superconducting Qubits, Trapped Ions, Neutral Atoms, Quantum Hardware, Instruction Set Architecture, 6-7 Topology, Directed Magic Routing, SevenQ-Diode, SQU-Diode, Adelic Simplicial Architecture, Resonance Processing Unit
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 (β→∞, tropical, classical), Forge (0<β<∞, real, statistical), Meld (β=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=diag(1,,eiπ/4) is the BIND opcode at j=21 — 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 argmaxg∈Gf(g) over an abelian group G has a C-MGE quantum counterpart C-MGE(f)=SPLITG⋅FLIP(f)⋅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, β→∞) recovers classical exhaustive search at cost O(N). "Warm-Shor" (Forge ISA, real β) gives Grover-like O(N) thermal concentration. Full Shor (Meld, β=it) achieves O(logN) 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: Ee∼Dσ[ei⟨ξ,e⟩]=e−∣ξ∣2σ2/2. 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 Hk complexity ladder (Paper 420) is realised by the trilogy: H0 (tree) = Origami, H1 (loop) = Forge/BPP, H2 (surface) = Meld/BQP. The T-gate is the H2 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 (⊕β 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
The state hidden subgroup problem (StateHSP) is a recent generalization of the hidden subgroup problem. We present an algorithm that solves the non-abelian StateHSP over N copies of the dihedral group of order 8 (the symmetries of a square). This algorithm is of interest for learning non-Pauli stabilizers, as well as related symmetries relevant for the problem of Hamiltonian spectroscopy. Our algorithm is polynomial in the number of samples and computational time, and requires only constant depth circuits. This result extends previous work on the abelian StateHSP and, as a special case, provides a solution for the ordinary hidden subgroup problem on this specific non-abelian group.