E0 · 7-qubit embedding
Clebsch–Gordan register.
2⁶ = 64 is too small. 2⁷ = 128 is the information-theoretic minimum. Three reserved codewords. The kernel sits in two contiguous coupled-basis blocks, so every E47 operation is a sandwich U_CG D₄₇ U_CG†.
qubits
7
min embedding
Hilbert
128
3 invalid
waste
2.3%
vs 75.6% naïve 9q
E47 blocks
2
10–34 · 90–111
Address map
product |m₁ m₂ m₃⟩ = |2, 0, 2⟩ · base-5 idx = 25·0 + 5·2 + 0 = 10
M₂ / M₅ multiplicity
Inside the kernel, E₄₇ ≅ (ℂ⁵ ⊗ ℂ⁵) ⊕ (ℂ² ⊗ ℂ¹¹). Collective SU(2) noise acts on the representation factors R₂ and R₅. The multiplicity spaces are noiseless for that declared algebra:
J = 2 · addresses 10–34
M₂ ≅ ℂ⁵ · 5-level qudit · log₂ 5 ≈ 2.32 bits
Casimir C = 6. SU(5) logical gates on the multiplicity.
J = 5 · addresses 90–111
M₅ ≅ ℂ² · binary logical qubit
Casimir C = 30. Internal gap ΔE = 24 from M₂.
This is not a claim of generic Knill–Laflamme QEC on all 47 dimensions.
Three-stage sandwich
Stage 1
Uncoupling U_CG† — product basis to coupled CG basis.
Stage 2
Block-diagonal D₄₇ on 10–34 and 90–111. Reflection uses +1 on E47, −1 on complement.
Stage 3
Recoupling U_CG back to the physical product basis.
R₄₇ = U_CG D₄₇ U_CG†
Generic 7-qubit unitary synthesis is O(4⁷) ~ 16k CNOTs — a design estimate, not a measured transpile. Structured CG networks are the intended research target. OpenPulse M₂ SU(5) schedule on transmons is listed at 31.68 µs, under 32% of typical T₁.