The Origami Compiler for Fault-Tolerant Quantum Computing: Exact Diagrammatic Compilation, Zero-Overhead CCZ, and the Regime Ladder for Magic-State Distillation
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. Magic-state distillation (MSD) is the dominant overhead cost in fault-tolerant quantum computing: each logical T or CCZ gate currently requires 100–1,000 physical qubits dedicated to distillation, consuming the majority of a quantum processor's resources. This paper introduces the Origami Instruction Set Architecture (Origami ISA) as a compiler layer for fault-tolerant quantum error correction — operating on existing lattice surgery hardware without modification. The five opcodes — FLIP, FLOP, SPLIT, SPLAT, TWIST — operate at the logical level and compute exact symbolic (rational or algebraic) amplitudes via the thermion library, enabling algebraic verification of gate correctness before any physical circuit is generated. Three contributions: 1. Frobenius cancellation: The Frobenius axiom (SPLIT(j) ∘ SPLAT(j)= id) is an automatic compiler optimisation that detects and removes redundant merge-boundary open/close pairs in lattice surgery protocols, reducing physical gate count by 29–67% on standard protocols. 2. Regime ladder: Every quantum error-correcting code is classified by its F-move amplitudes. Rung-0 codes (stabiliser codes, surface codes) require MSD for non-Clifford gates. Rung-1 codes ([[15,1,3]] Reed-Muller, colour codes) support transversal non-Clifford gates with zero MSD overhead. The ladder predicts which hardware can eliminate MSD and which cannot. 3. Zero-MSD CCZ protocol: The SPLIT;FLOP;FLOP;CCZ;FLOP⁻¹;FLOP⁻¹;SPLAT protocol implements a universal CCZ gate on the combined [[7,1,3]] + 1531-Anvil hardware without any magic-state distillation. Machine-verified correctness via the Frobenius axiom and Pentagon identity, proved by exact symbolic arithmetic. The Pentagon identity holds exactly for all five codes examined — proving by exact symbolic arithmetic rather than numerical simulation. Keywords Origami ISA, Fault-Tolerant Quantum Computing, Magic State Distillation, Lattice Surgery, Frobenius Axiom, Pentagon Identity, Regime Ladder, CCZ Gate, Transversal Gates, Reed-Muller Code, [[15,1,3]] Code, Steane Code, [[7,1,3]] Code, 1531-Anvil, FLIP Opcode, FLOP Opcode, SPLIT Opcode, SPLAT Opcode, Quantum Compiler, Diagrammatic Calculus, Exact Arithmetic, Thermion Library, F-Move Amplitude, Colour Code, Surface Code, Quantum Error Correction, Zero Overhead
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. Magic-state distillation (MSD) is the dominant overhead cost in fault-tolerant quantum computing: each logical T or CCZ gate currently requires 100–1,000 physical qubits dedicated to distillation, consuming the majority of a quantum processor's resources. This paper introduces the Origami Instruction Set Architecture (Origami ISA) as a compiler layer for fault-tolerant quantum error correction — operating on existing lattice surgery hardware without modification. The five opcodes — FLIP, FLOP, SPLIT, SPLAT, TWIST — operate at the logical level and compute exact symbolic (rational or algebraic) amplitudes via the thermion library, enabling algebraic verification of gate correctness before any physical circuit is generated. Three contributions: 1. Frobenius cancellation: The Frobenius axiom (SPLIT(j) ∘ SPLAT(j)= id) is an automatic compiler optimisation that detects and removes redundant merge-boundary open/close pairs in lattice surgery protocols, reducing physical gate count by 29–67% on standard protocols. 2. Regime ladder: Every quantum error-correcting code is classified by its F-move amplitudes. Rung-0 codes (stabiliser codes, surface codes) require MSD for non-Clifford gates. Rung-1 codes ([[15,1,3]] Reed-Muller, colour codes) support transversal non-Clifford gates with zero MSD overhead. The ladder predicts which hardware can eliminate MSD and which cannot. 3. Zero-MSD CCZ protocol: The SPLIT;FLOP;FLOP;CCZ;FLOP⁻¹;FLOP⁻¹;SPLAT protocol implements a universal CCZ gate on the combined [[7,1,3]] + 1531-Anvil hardware without any magic-state distillation. Machine-verified correctness via the Frobenius axiom and Pentagon identity, proved by exact symbolic arithmetic. The Pentagon identity holds exactly for all five codes examined — proving by exact symbolic arithmetic rather than numerical simulation. Keywords Origami ISA, Fault-Tolerant Quantum Computing, Magic State Distillation, Lattice Surgery, Frobenius Axiom, Pentagon Identity, Regime Ladder, CCZ Gate, Transversal Gates, Reed-Muller Code, [[15,1,3]] Code, Steane Code, [[7,1,3]] Code, 1531-Anvil, FLIP Opcode, FLOP Opcode, SPLIT Opcode, SPLAT Opcode, Quantum Compiler, Diagrammatic Calculus, Exact Arithmetic, Thermion Library, F-Move Amplitude, Colour Code, Surface Code, Quantum Error Correction, Zero Overhead
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 Origami ISA — five opcodes (SPLIT, SPLAT, TWIST, FLIP, FLOP) corresponding to Pachner moves on a triangulated simplicial complex — is already a combinator library, but without a machine-checkable type system. This paper makes the type system explicit, yielding three progressively expressive type systems and one key theorem: associamancy is a compile-time type error. Type System 1 — Simple types. Wire types are irreducible representation labels Rep(G) for a gauge group G. For the Origami ISA: G = SU(2), types are spins j ∈ {0, ½, 1, …}. For the 731 ISA: G = G₂, types are Dynkin labels (p,q). For the financial Pacioli ISA: G = (ℝ₊, ×), types are currencies. The Pentagon identity d² = 0 becomes a typing rule: SPLAT(SPLIT(x)) must return the trivial representation 1. Any circuit producing a non-trivial output is ill-typed — this is the type-level expression of the leakage criterion introduced in Paper 411. Type System 2 — Linear types. Every wire is usable exactly once, encoding the quantum no-cloning theorem. SPLIT is the unique duplication primitive (1 → 4 wires, the Pachner move); FLOP is the unique elimination primitive (trace / Born rule). The category-theoretic model is a symmetric monoidal category extending the ZX-calculus to Rep(G)-typed wires over arbitrary compact groups G. Type System 3 — Dependent types. The output type of SPLIT depends on the input type via the Clebsch-Gordan decomposition: SPLIT_CG : (j : Rep(G)) → Σ(j₁ j₂ j₃ j₄ : Rep(G)). CG(j; j₁, j₂, j₃, j₄) ≠ 0 The fibred-category model (Grothendieck fibration over Rep(G)) captures this: the Schur boundary ν₂ ≠ 0 selects the associative subcategory (Origami ISA, regime 2) from the full category (731 ISA, regime 3). The Schur boundary as a TypeError. The Frobenius-Schur indicator ν₂ ∈ {−1, 0, +1} partitions wire types into two subtypes: AssocWire (ν₂ ≠ 0, Origami ISA) and FullWire (ν₂ = 0 allowed, 731 ISA). The SPIN opcode — the generator of triality, the order-3 outer automorphism of Spin(8) with G₂ = Fix(triality) — has type FullWire → FullWire with the constraint ν₂ = 0. Applying SPIN to an AssocWire raises a TypeError at circuit construction time, before any hardware is consulted. Associamancy (Paper 407) is no longer only a resource-theoretic concept — it is a compile-time type mismatch. The three-level resource hierarchy as a type tower: Level Resource Wire subtype Protection 0 Stabiliser Any Rep(SU(2)) None 1 Standard magic AssocWire (ν₂ ≠ 0) None 2 Associamancy FullWire (ν₂ = 0) Discrete Berry phase (W(G₂), order 12) The Pacioli Combinator Library as the financial instance. The PCL (Paper 306) — typed combinators flow, sequence, choose, fold — is the financial instance of this DSL with G = (ℝ₊, ×). The Pacioli identity (every debit has a credit) is d² = 0 at the financial instance. The same typed circuit compiles to quantum hardware, financial risk models, or nuclear spectroscopy; only the gauge group G and the wire type interpretation change. Python prototype included: ISACircuit class with runtime type checking; spin() raises TypeError if ν₂ ≠ 0; pentagon_check() validates d² = 0 at construction time. Keywords: Origami ISA, typed DSL, linear types, dependent types, Schur boundary, associamancy, no-cloning, Pentagon identity, Clebsch-Gordan, Pacioli combinator library, symmetric monoidal category, ZX-calculus, G₂, triality, type theory, quantum circuits
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 articulate a structural parallel between the MIP*=RE theorem and the Origami Instruction Set Architecture (ISA). The algebraic rigidity of entangled-prover verification — exact-or-fail Pauli basis tests, low-degree tests, Mermin Magic Square constraints — corresponds precisely to the Frobenius axiom SPLIT∘SPLAT=1, the Pentagon identity, and the seven-point Fano plane PG(2,2) in the ISA framework. Six numerical experiments confirm the parallels: Pentagon unitarity holds exactly for all j ∈ {1/2,...,15/2}; Frobenius fidelity degrades as (1-p)^{2(2j+1)} under depolarising noise; the GHZ stabiliser group forms a perfect quantum Fano strategy satisfying all seven constraints simultaneously while no classical strategy can; G₂ non-associativity produces a sharp excluded-volume step function above j=7/2. This version adds a synthesis remark showing that the numbers 3 and 7 — the two dimensions where the cross product exists, gyroscopes are stable, and choreographic orbits can close — are not two separate coincidences but the same theorem: the Fano plane PG(2,F₂) has 7 points and 3 points per line because these are the imaginary unit counts of the octonions O and quaternions H respectively, forced by the unique projective plane of order q=2 over the field F₂. The broken Fano (6/7 architecture) and the magic/non-associativity conjecture are restated in this unified language. The practical implication is a conjecture that fault-tolerant quantum computing overhead is governed by the regime of the code's F-move amplitudes, with MSD-free computation possible at rung 1 and higher. Keywords MIP* Equals RE, Interactive Proof Systems, Quantum Entanglement, Pentagon Identity, Frobenius Axiom, Fano Plane, PG(2,2), GHZ Stabiliser, Origami ISA, Wigner 6j Symbol, Fault-Tolerant Quantum Computing, Magic State Distillation, Regime Ladder, Complexity Theory, Quantum Prover, Local Pauli Test, Low-Degree Test, G₂ Non-Associativity, Excluded Volume, Depolarising Noise, Rung-1, Quantum Verification, Adelic Simplicial Architecture
Zero-level CCZ Distillation
Magic state distillation is a key component of fault-tolerant quantum computation, as it enables the implementation of non-Clifford gates such as the T gate and the CCZ gate via gate teleportation. However, conventional distillation protocols require a large number of logical qubits and introduce substantial spatial and temporal overhead, posing a significant bottleneck for scalable fault-tolerant quantum computation. In this work, we propose a zero-level distillation protocol that efficiently generates a high-fidelity logical CCZ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal T/T† operation of the [[8,3,2]] code to fault-tolerantly encode the state CCZ∣+++⟩, which is subsequently teleported to three surface-code logical qubits via lattice surgery. To enable teleportation between codes with different distances, we introduce adaptively initialized teleportation (AIT), a tailored initialization procedure for the surface code. Numerical simulations demonstrate that the logical error rate scales as pL≃300×p2 with respect to the physical error rate p. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at p=10−3 and p=10−4, respectively, compared to conventional seven-T-gate approaches. The distillation circuit requires only 22 physical qubits, 3 logical qubits, and a circuit depth of 24, reducing the space-time overhead by a factor of approximately 5-10 compared to previous methods. This result highlights the practicality of CCZ-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional T-state generation.
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 seven opcodes of the Origami instruction-set architecture (ISA) — FLOP, SPLIT, TWIST, LABEL, BIND, FLIP, SPLAT — play structurally identical roles across every exactly solvable quantum model with a phase transition. This paper establishes a universal dictionary: FLOP is always the fermionisation map (Jordan–Wigner / slave-boson / Majorana representation); SPLIT is always the diagonalisation (Bogoliubov / Bethe rapidity / stabiliser expansion); TWIST is always the gap check (Berry phase / Chern number / spectral gap); LABEL is always the sector projection (gauge fixing / stabiliser projection / Bethe vacuum). BIND — the three-body frog vertex — is the unique opcode whose presence or absence determines whether a phase supports Abelian or non-Abelian anyons. Two theorems are proved. The BIND = Non-Abelian theorem: a gapped topological phase has non-Abelian anyonic order if and only if its minimal ISA programme contains BIND. The Universal Phase Boundary theorem: for any model in the SU(2)q family at q = exp(iπβ), the quantum phase transition is a TWIST failure at β = 1/2, where the quantum dimension d{1/2}(β) = 2cos(πβ) = 0 exactly. A master table across twelve models — toric code, Kitaev honeycomb, TFIM, XXZ chain, string nets, quantum doubles, SYK, MBQC, ZX-calculus, and others — makes the dictionary explicit. The Origami ISA is the universal language in which condensed-matter physicists, quantum-information theorists, and representation theorists describe the same computation under different names. Keywords instruction set architecture, exactly solvable models, topological order, non-Abelian anyons, quantum phase transitions, Origami ISA, Jordan–Wigner transformation, Bogoliubov transformation, Bethe ansatz, toric code, Kitaev honeycomb, XXZ chain, string-net condensation, quantum doubles, SYK model, MBQC, ZX-calculus, Frobenius algebra, quantum groups, Berry phase, BIND theorem, β-ladder, tensor networks, categorical quantum mechanics
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
Scalable quantum computation is expected to rely on fault-tolerant quantum computation (FTQC), in which quantum error correction (QEC) suppresses physical errors sufficiently to support reliable logical operations. This requires quantum compilation to move beyond general-purpose circuit optimization toward encoding-aware and protocol-structured compilation across the full stack of fault-tolerant quantum computers. Beyond circuit synthesis and hardware mapping, an FTQC compiler must lower algorithm-level operations into the logical gate set supported by the chosen code, coordinate encoded data and ancilla resources, realize logical operations together with repeated syndrome extraction under hardware constraints, and provide the resulting measurement stream to real-time decoding.
This survey presents a full-stack view of compiler design for QEC-protected quantum computation. We organize existing work into three interacting layers: logical-level QEC compilation, physical-level QEC realization, and decoder runtime integration. At the logical level, we review surface-code lattice-surgery compilers, beyond-surface-code code-surgery frameworks including emerging qLDPC approaches, and compilation support for non-Clifford operations such as magic-state distillation and code switching. At the physical level, we survey hardware-aware QEC realization on superconducting, trapped-ion, and neutral-atom platforms. We further examine decoder models, real-time decoding systems, and frame-management mechanisms that close the feedback loop during fault-tolerant execution. Finally, we identify open challenges in cross-layer optimization, qLDPC compilation, compiler-decoder co-design, runtime adaptivity, and the development of integrated and benchmarkable FTQC compilation stacks. An actively maintained paper list is available at: github.com/chenghongz/QEC-compiler-design.