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

In the standard circuit model, elementary gates are defined relative to a chosen tensor factorization and are therefore extrinsic to the ambient group $U(2^n)$. Writing $N=2^n$, we introduce an \emph{intrinsic descriptor layer} in $U(N)$ by declaring as primitive the motions inside faithful embedded copies of $SU(2)$ (phase-free), together with a phase-inclusive $U(2)$ variant. We describe the embedding landscape $\Emb(SU(2),U(N))$ as a finite union of $U(N)$-homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers, and we isolate a canonical \emph{two-level sector} parameterized by $\Gr_2(\C^N)$ up to a $PSU(2)$ gauge. Equipping $U(N)$ with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic, yielding a variational characterization of elementary motions via minimal-norm logarithms. On the constructive side, we prove phase-free universality in $SU(N)$ from two-level primitives using QR/Givens factorizations together with explicit diagonal generation, and we obtain full universality in $U(N)$ by explicit abelian phase bookkeeping (equivalently, via the $U(2)$ two-level dictionary). Finally, we formalize a modular finite-alphabet compilation interface: any approximation routine in $SU(2)$ (e.g.\ Solovay--Kitaev) can be lifted through two-level embeddings to yield $U(N)$-level synthesis with global operator-norm error control.