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 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
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 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
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 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.
Short note with the circuit, an independent verifier and the search records. For ancilla-free Clifford preparation of the logical zero |0_G> = 2^{-11/2} Σ_{c ∈ C^⊥} |c> of the [[23,1,7]] quantum Golay code (Open Problem Zoo op_c847b635f53b0812, after Paetznick and Reichardt 2012), the best documented circuits were 45 two-qubit gates at two-qubit depth 7 and 44 gates at depth 8 (Doherty et al., arXiv:2603.17743). We exhibit an explicit circuit with 44 two-qubit gates (all controlled-Z, i.e. CNOT up to free single-qubit Cliffords) in six disjoint two-qubit layers of sizes 9, 10, 7, 6, 7, 5, so g_G(D) ≤ 44 for every D ≥ 6; it dominates both previous Pareto points. The circuit is given in the cyclic coordinates of the Golay code (generator polynomial 1+x^2+x^4+x^5+x^6+x^10+x^11) and verified two independent ways: a stabilizer-tableau proof (the output stabilizer group is CSS with all + signs, its Z-code is a binary [23,12,7] code, hence the Golay code by uniqueness, and an explicit wire permutation is exhibited) and a full 2^23-amplitude statevector overlap of 1.0 with the target. The circuit was found by an autonomous AI research agent (the depositor's EVO discovery harness, Claude Sonnet 5.5 proposer) by a local search started from the published 44-gate depth-8 circuit: a bit-packed C tableau simulator, a cost equal to the number of qubits left entangled when the circuit is run backwards from the target, and a random walk among 44-gate circuits that penalises gates in layers beyond the target depth. The same searches did not find 43 gates at any depth or 44 gates at depth 5 (about 10^10 and 10^9 evaluations respectively; these are negative search outcomes, not bounds). Claim boundary: an upper bound g_G(6) ≤ 44 witnessed by an explicit, machine-checkable circuit; no optimality or priority claim. Files: the note (PDF + LaTeX), the circuit in three equivalent JSON forms (agent wire labels; cyclic coordinates; cyclic coordinates with CNOT as the only two-qubit gate), the verifier and relabeling finder, the agent's notes and search code, the task text, SHA-256 sums.
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup H of a group G must be determined from a quantum state ψ over G that is uniformly supported on a left coset of H. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of ψ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups H of the q-hedral groups, i.e., semidirect products Zq⋉Zp, where q∣(p−1), and in particular the affine groups Ap, can be information-theoretically reconstructed using the strong standard method. Moreover, if ∣H∣=p/polylog(p), these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We compare our algorithms to two weaker methods that have been discussed in the literature—the “forgetful” abelian method, and measurement in a random basis—and show that both of these are weaker than the strong standard method. Thus, at least for some families of groups, it is crucial to use the full power of representation theory and nonabelian Fourier analysis, namely, to measure the high-dimensional representations in an adapted basis that respects the group's subgroup structure. We apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems, generalizing the work of van Dam, Hallgren, and Ip on shifts of multiplicative characters. Finally, we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently.