Prior work by the authors [1] showed that the 24 vertices of the 24-cell, as unit quaternions, form the binarytetrahedral group 2T, that a controlled probe lifts the phase-blind projective group 2T/{±1} to all 24 gates, and thatthe optimal single-shot measurement discriminating them compiles, gate by gate, to an explicit RY/RZ/X/CNOTcircuit. That compilation was symmetry-blind: a textbook Givens synthesis of a generic 32×32 unitary, emitting25,210 CNOTs. Here we ask what the group structure of the input buys back. We report four findings. (A) The 24controlled-probe states span only a rank-3 subspace, so a reader need be correct only there; restricting thesynthesis to that support cuts the emitted CNOT count from 25,210 to 5,980 (4.2×) with the same single-shotarchitecture and the identical optimum. (B) The optimal POVM is exactly 2T-covariant: its 24 elements are onegroup orbit of a single seed, and the analysis map is a genuine tight-frame isometry that intertwines the supportrepresentation with the regular representation of 2T. (C) Consequently the reader factors as a data-independent2T group-Fourier transform followed by a placement whose data-dependent part is tiny (72 CNOT); the rank-3signal occupies just 5 of the 24 Fourier lines, which is the structural reason (A) works. (D, honest negative) Acheap synthesis of that group-Fourier transform is not achieved: the naive tensor product of the C3 and Q8transforms fails to block-diagonalize because 2T = Q8 : C3 is a crossed product, and the missing twiddle absorbsall the cost. We make no quantum-advantage claim; every number is a real emitted gate count or a verified matrixidentity from exact, numpy-only statevector simulation. ※ The license for codes is AGPL3.0-or-later
Prior work by the authors [1] showed that the 24 vertices of the 24-cell, as unit quaternions, form the binarytetrahedral group 2T, that a controlled probe lifts the phase-blind projective group 2T/{±1} to all 24 gates, and thatthe optimal single-shot measurement discriminating them compiles, gate by gate, to an explicit RY/RZ/X/CNOTcircuit. That compilation was symmetry-blind: a textbook Givens synthesis of a generic 32×32 unitary, emitting25,210 CNOTs. Here we ask what the group structure of the input buys back. We report four findings. (A) The 24controlled-probe states span only a rank-3 subspace, so a reader need be correct only there; restricting thesynthesis to that support cuts the emitted CNOT count from 25,210 to 5,980 (4.2×) with the same single-shotarchitecture and the identical optimum. (B) The optimal POVM is exactly 2T-covariant: its 24 elements are onegroup orbit of a single seed, and the analysis map is a genuine tight-frame isometry that intertwines the supportrepresentation with the regular representation of 2T. (C) Consequently the reader factors as a data-independent2T group-Fourier transform followed by a placement whose data-dependent part is tiny (72 CNOT); the rank-3signal occupies just 5 of the 24 Fourier lines, which is the structural reason (A) works. (D, honest negative) Acheap synthesis of that group-Fourier transform is not achieved: the naive tensor product of the C3 and Q8transforms fails to block-diagonalize because 2T = Q8 : C3 is a crossed product, and the missing twiddle absorbsall the cost. We make no quantum-advantage claim; every number is a real emitted gate count or a verified matrixidentity from exact, numpy-only statevector simulation. ※ The license for codes is AGPL3.0-or-later
The parent paper [1] reduced the circuit synthesis of the optimal 24-outcome reader for the binarytetrahedral group 2T to a single fixed operator, the 2T group Fourier transform, and identified its dominantcost as FQ8, the 8×8 group Fourier transform of the quaternion group Q8. The companion addendum [2]reported, as an honest negative, that on the parent paper's generic Givens-based emitter astructurally-built FQ8 still costs 218-234 CNOT -- far above the O(101) structural estimate -- and framed thegap as a binary-embedding and generic-decomposition tax: an emitter artifact rather than arepresentation-theoretic obstruction. This note discharges that claim with verified circuits. Using only thestructure of FQ8 -- the plus/minus pairing induced by the center Z2 of Q8, which lets a single Hadamardseparate the character sector from the two-dimensional-representation sector -- we build an exactfactorization in which the sole entangling object is a single controlled two-qubit gate Cq2[Wr]. We thenshow that Wr has Cartan (KAK) coordinates (π/4, 0, 0), the CNOT class, so its two-qubit nonlocal core isone CNOT. Three constructions of FQ8 are emitted and verified equal to FQ8 up to global phase atmachine precision, giving a monotone progression of 234 → 33 → 23 emitted CNOT. Every number belowis a real emitted, verified circuit, not a cost-model estimate. We are explicit about what is not achieved: thesingle-digit floor implied by "one CNOT core" is not reached, because lifting Wr to its controlled form costsone Toffoli plus controlled-local flanks (14 CNOT as emitted); reducing this further is a compiler task, not anew representation-theoretic result. The result establishes that the 200-CNOT wall was largely an emitterartifact. ※ The license for codes is AGPL3.0-or-later
The parent paper [1] reduced the circuit synthesis of the optimal 24-outcome reader for the binary tetrahedralgroup 2T to the synthesis of a single fixed operator: the 2T group Fourier transform F, whichblock-diagonalizes the regular representation into irreducible blocks [1,1,1, 2,2,2, 2,2,2, 3,3,3]. That paperreported, as an honest negative (its Result D), that the naive tensor product FC3 ⊗ FQ8 fails to realize F,because 2T = Q8 : C3 is a crossed (semidirect) product, and it named the missing piece — the twiddle fromthe Clifford theory of Q8 : C3 — as a remaining construction. Here we construct it. We show (i) the twiddle hasan exact closed form: in coset-times-fiber coordinates, each two-dimensional-sector column of F equals (ωmM0)t applied to the Q8-Fourier column, where M0 is the intertwiner of the two-dimensional irrep ρ with itsC3-conjugate (rebuild residual 1.7 × 10-16); (ii) a semidirect-product Fourier obstruction: nodirect-product-form two-factor split of F exists, because the C3-action moves the Fourier dual of Q8 nontrivially(M0 ≠ I) — stated and verified as a direct-versus-semidirect contrast, not as an observed residual; and (iii) adirect measurement of the binary-embedding tax: the abstract group-FFT gate count is O(101), but ageneric-Givens emitter on the 24-into-32 five-qubit embedding cannot realize it, so the O(102) target isreachable in principle yet unreached with a structure-blind emitter. The line's firmly established compressionremains the parent paper's Lever-1 (25,210 → 5,980 CNOT, 4.2×). No quantum-advantage claim is madeanywhere. All results are exact statevector-level numpy computations at machine precision. ※The license for codes is AGPL3.0-or-later
The Watabe-Claude Method reads a signal's structure by assigning 4D-embeddedwindows to the 24 vertices of the 24-cell and decomposing the occupancy with thegroup B4. This note takes the 24-cell the other way -- into a qubit. Its 24vertices, as unit quaternions, are exactly the binary tetrahedral group 2T, afinite nonabelian subgroup of SU(2); under the quaternion-to-SU(2) map they are24 distinct single-qubit gates. Three exact facts, all verified by classical simulation of one or two qubits(no external quantum or group-theory libraries): 1. Foundation. The 24 vertices form 2T (order 24): closed Cayley table, Latin square, 24-cell inner-product signature {-1,-1/2,0,1/2,1}. As SU(2) matrices they are 24 distinct, nonabelian gates containing the Pauli set. 2. An honest negative. Encoding the vertex assignment for a quantum optimiser is a dead end: the 5-qubit binary QUBO has 31 terms (vs 300 one-hot) but up to five-body couplings, and a shallow QAOA recovers the right vertex only 3-10% of the time -- the assignment is a classical argmax with no role for a quantum optimiser. 3. The finding. Applied to a single probe qubit, the 24 gates collapse to 12 distinct outputs, because 2T's center {+-1} is exactly the global phase a lone qubit cannot observe (U(-q)=-U(q) verified for all 24); the observable group is the projective image 2T/{+-1}, the order-12 tetrahedral rotation group. A controlled application makes the center's phase relative and separates all 24. Optimal single-shot (square-root) discrimination succeeds 16.7% over the 12 classes and 12.1% over all 24 -- 2.0x and 2.9x the random baselines, ratios fixed by the 24-cell geometry. The through-line with the classical WCM series: there, B4 reads which symmetrychannel carries a signal's structure; here, the same polytope's group structuregoverns which distinctions a quantum measurement can make. No quantum speedup oradvantage of any kind is claimed; the contribution is an exact correspondencebetween a polytope's group structure and the observability limits of a qubit.Self-audit: 12 findings, all NUMERICAL / NEGATIVE / STRUCTURAL, none overclaimed. v1.0.1 (2026-08-27): Adds an explicit, end-to-end-verified measurement circuit. The optimal square-root discrimination is certified minimum-error (Yuen-Kennedy-Lax conditions plus an independent iteration from random measurements), realised as an explicit Naimark unitary, embedded in a power-of-two qubit register, and compiled to an atomic gate list of RY, RZ, X, and CNOT only -- verified end to end (every gate one of those four types, the product reproducing the target measurement within floating-point accumulation; ~2e-11 over the ~84,000-gate 24-way circuit). The lone-qubit collapse is sharpened to an orbit hierarchy (24 -> 12 generic -> 6 symmetry-axis), naming the measure-zero exceptional probes (including the computational basis). Terminology and scope clarifications throughout (B4 vs 2T; central quotient; coherent-control assumption; honest symmetry-blind CNOT upper bounds 25,210 / 2,022). No quantum speedup is claimed. Status: preprint, not peer-reviewed. This is an independent research note; its revisions were shaped by critique from an AI collaborator, which is not a substitute for formal peer review. Text CC-BY-4.0; code AGPL-3.0-or-later.
The group Fourier transform of the binary tetrahedral group 2T = SL(2,3) = Q8 : C3 resists a naive tensorfactorisation F3 ⊗ FQ8 because the group is a genuine semidirect (crossed) product. We reconstruct the correctassembly explicitly. Writing the transform as W = U2 · (I3 ⊗ FQ8)†, the residual operator U2 is not a globalproduct but a direct sum over the C3-orbits of the Q8 dual: a trivial-character sector, a sign-character orbitsector, and a two-dimensional-irrep sector, of dimensions 3, 9 and 12. We give the representation identitybehind the shorthand 24 = 3 ⊕ 9 ⊕ (2×6), verified to machine precision by exact block-diagonalisation of theregular representation (maximum off-block 1.35×10−15). Two structural facts organise the synthesis. First, thesign sector maps a monomial action to a monomial target, so its intertwiner can be a pure basis permutation;the two-dimensional-irrep sector maps a monomial action to a non-monomial target, so its intertwiner provablycannot be permutation-diagonal. Second, the twelve-dimensional sector factors as ρ0 ⊗ D6 with D6 = D3 ⊗ I2,isolating a single common two-dimensional representation factor. We stress the distinction betweenrepresentation structure and circuit cost: the factorisation is a statement about the assembly, not yet a verifiedentangling-gate count. A self-audit table records which invariant refuted each earlier, incorrect decomposition. ※ The license for codes is AGPL3.0-or -later
Hamiltonian simulation is a core quantum workload, underpinning variational quantum algorithms and Trotterized time evolution. Such programs are expressed as Pauli exponential sequences, exhibiting structural patterns that are highly amenable to high-level synthesis and optimization. Existing compilers, however, fail to fully unlock the optimization potential of their global algebraic structure, even when employing advanced graph- or tableau-based methods.
We present Symphony, a holistic compilation approach built on the binary symplectic form (BSF) representation of Pauli strings. Unlike prior group-wise BSF simplification and path-based Pauli network synthesis, Symphony applies generalized controlled-Pauli Clifford transformations directly to a global BSF tableau, progressively reducing active Pauli rows while adaptively emitting eligible two-qubit blocks in addition to single-qubit rotations in a forward Clifford frame. Following algebraic simplification, Symphony performs a causality-preserving block rescheduling heuristic that respects frame-induced dependencies while exposing extensive two-qubit block parallelism opportunities. This streamlined compilation style comprehensively exploits simultaneous simplification and commutativity opportunities, achieving efficient global optimization without relying on computationally expensive heuristics or long-horizon searches. Across the generic Hamiltonian simulation benchmarks in HamLib, Symphony achieves average reductions of 59% in two-qubit gate count and 91% in circuit depth. It Pareto-dominates prior state-of-the-art compilers, requiring 1.17--1.58× fewer two-qubit gates and especially shrinking two-qubit circuit depth by a substantial factor of 1.87--5.67× on average.
This work explores the composition of Microsoft's 4D geometric error-correcting codes — built on the 24-cell honeycomb cellulation — into higher-dimensional logical structures targeting 24D stability. While 4D codes have demonstrated 1,000-fold error rate reduction and single-shot correction on neutral atom arrays, this work asks how these codes compose when stacked or tensored into 8D, 12D, or 24D logical architectures. We investigate the stabilizer structure under composition, whether single-shot correctability survives dimensional lifting, and the mapping of composed code geometries onto neutral atom hardware with all-to-all connectivity. This bridges the gap between demonstrated 4D fault-tolerant codes and the theoretical framework for topological codes beyond dimension 2. Update, 30 July 2026 — constructive resolution. The distance question raised here is now settled by direct construction in 10.5281/zenodo.21698294, which builds the 4D toric code from its cubical chain complex and obtains [[96,6,4]] at L=2: k=6 exactly by GF(2) rank, and distance 4 proven exhaustively over every error pattern of weight ≤4. Distance 8 is therefore not obtainable from the 4D toric construction. That bounds the 4D toric family only and does not establish that no [[96,6,8]] code exists in another family — the quantum Singleton bound permits it comfortably at these parameters. An earlier wording of this note, added and corrected on 30 July 2026, overstated the scope. That work was carried out independently, from first principles, and reaches the same conclusion. Erratum, 2026-07-31 — the novelty claim is withdrawn, and what replaces it. The deposited file states that "nobody has published on composing 4D geometric codes into higher-dimensional logical structures" and that the bridge paper "has not been written." That is false. Composing chain-complex CSS codes is the homological (tensor) product of chain complexes, and the cellular chain complex of a product complex is the tensor product of the factors', so the composition of two 4D cubic toric codes is the 8D cubic toric code — a published family, and the subject of this record's own reference #9. This was checked, not assumed: an explicitly composed 8D code and an independently built 8D cubic toric code agree at all 8 grades on shape, nonzero count, both check-weight spectra and exact GF(2) rank. The composition constructs no new code. What replaces the claim is a quantitative answer to the question the record was actually reaching for. For the balanced hypercubic toric code at middle grade, n/k = L^D exactly and independently of D — verified in 8 of 8 computed cases, with the qubit count a combinatorial count and the logical count a measured pair of GF(2) ranks, up to n = 459,270 where both ranks were measured directly. Composing therefore changes how many logical qubits sit in one block and changes nothing else. Head to head at the same distance 81, the 8D composite [[459270, 70, d ≤ 81]] and the plain 4D code [[39366, 6, d ≤ 81]] both spend 6,561 physical qubits per logical qubit, to the digit — with the 8D version paying weight-10 stabilizers where the 4D pays weight-6. Distance does not rescue it either: by Künneth the distance of a product is bounded by the minimum over grade splittings, not the product of the factors' distances, so each added dimension is a chance to lose distance — the 5D product of [[486,6,9]] at grade 1 is [[1215, 5, 3]], proved exactly. On single-shot correction: there is nothing to lift. D = 4 at q = 2 already has metachecks in both sectors (ranks 80 and 80 for [[486,6,9]]), so 8D, 12D and 24D add no sector that was not already covered — and lifting can lose it, the 5D product at grade 1 having no X-sector metachecks at all (rank d₀ = 0). On hardware: all-to-all connectivity is real but was never the binding constraint. The smallest possible balanced 24D hypercubic toric code needs 4.5 × 10¹³ physical qubits, 1.5 × 10¹⁰ times the 3,000-atom array cited above. What stands: the homological product is the correct composition operation and the framework is well-posed; composition genuinely does raise dimension and does buy more logical qubits per block (6 → 70 from 4D to 8D); the metacheck question was the right one to ask; and the reference library is accurate. Two limits on the erratum itself: all computation uses the cubic 4-torus code [[486,6,9]], not Microsoft's 24-cell honeycomb code, whose parity-check matrices were never obtained — so nothing here is a statement about that code; and no exact distance is claimed at n ≥ 2430, the 8D figures being certified upper bounds with no lower bound proved. The 1,000-fold error-rate reduction and the neutral-atom hardware results were not tested and are neither confirmed nor refuted. Full computation, code and logs: 10.5281/zenodo.20808710 and branch writeup at /home/voodoo/branch_work/18779369_composition/RESULT.md. Note: the source file QEC_Neutral_Atom_24D_Research.md itself still contains the withdrawn claim verbatim (a published file cannot be edited in place) — this erratum is the authoritative correction until a new version replaces the file text.