Elementary Quantum Gates from Lie Group Embeddings in U\(2ⁿ\): Geometry, Universality, and Discretization

In the standard circuit model, elementary gates are specified relative to a chosen tensor factorization and are therefore extrinsic to the ambient group U\(2ⁿ\). Writing N=2ⁿ, we introduce an intrinsic descriptor layer in U(N) by declaring as primitive the motions inside faithful embedded copies of SU(2), leading to the phase-free dictionary mathcal{G}^SU_elem(n)=bigcup_{φinEmb(SU(2),U(N))}φ(SU(2)), and we also discuss the phase-inclusive U(2) variant. We show that Emb(SU(2),U(N)) decomposes into finitely many U(N)-homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers; the canonical two-level sector is organized by Gr₂\(C^N\) up to a PSU(2) gauge. Equipping U(N) with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic. Using two-level QR/Givens factorization together with an explicit generation of diagonal tori by two-level phase rotations, we prove phase-free universality langlemathcal{G}^SU_2lvl(n)rangle=SU(N) and hence langlemathcal{G}^SU_elem(n)rangle=SU(N). Full universality in U(N) follows by adjoining the abelian diagonal/global U(1) factors (equivalently, by passing to the U(2) two-level dictionary). Finally, we record a modular finite-alphabet interface by lifting Solovay--Kitaev approximation in SU(2) through two-level embeddings.