Symmetry-Adapted Synthesis of the 24-Cell Quantum Reader

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