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

Antonio Falco, Daniela Falco-Pomares, Hermann G. Matthies

TL;DR

To address intrinsic elementarity inside $U(2^n)$, the authors model a logical qubit as a faithful embedding of $SU(2)$ into $U(N)$ and organize embeddings by isotypic multiplicities, yielding a Grassmannian-based two-level sector. They prove phase-free universality: $SU(N)$ is generated by two-level $SU(2)$ gates via a two-level QR/Givens factorization and diagonal torus generation by two-level phase rotations; full universality in $U(N)$ follows after incorporating diagonal/global phases. They then develop a modular discretization pipeline by lifting Solovay–Kitaev approximations in $SU(2)$ through two-level embeddings to obtain operator-norm error control in $U(N)$. The geometry is anchored by total-geodesy of embedded subgroups, giving a variational interpretation where minimal-energy implementations are constant-speed geodesics generated by minimal-norm logarithms; together, these yield a robust, intrinsic, and discretizable gate framework independent of a fixed tensor factorization.

Abstract

In the standard circuit model, elementary gates are specified relative to a chosen tensor factorization and are therefore extrinsic to the ambient group $U(2^n)$. Writing $N=2^n$, 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 $\mathcal{G}^{SU}_{\mathrm{elem}}(n)=\bigcup_{φ\in\Emb(SU(2),U(N))}φ(SU(2))$, 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 $\Gr_2(\C^N)$ up to a $PSU(2)$ gauge. Equipping $U(N)$ with the Hilbert--Schmidt bi-invariant metric, each embedded subgroup is totally geodesic. Using two-level QR/Givens factorization together with an explicit generation of diagonal tori by two-level phase rotations, we prove phase-free universality $\langle\mathcal{G}^{SU}_{\mathrm{2lvl}}(n)\rangle=SU(N)$ and hence $\langle\mathcal{G}^{SU}_{\mathrm{elem}}(n)\rangle=SU(N)$. 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.

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

TL;DR

To address intrinsic elementarity inside , the authors model a logical qubit as a faithful embedding of into and organize embeddings by isotypic multiplicities, yielding a Grassmannian-based two-level sector. They prove phase-free universality: is generated by two-level gates via a two-level QR/Givens factorization and diagonal torus generation by two-level phase rotations; full universality in follows after incorporating diagonal/global phases. They then develop a modular discretization pipeline by lifting Solovay–Kitaev approximations in through two-level embeddings to obtain operator-norm error control in . The geometry is anchored by total-geodesy of embedded subgroups, giving a variational interpretation where minimal-energy implementations are constant-speed geodesics generated by minimal-norm logarithms; together, these yield a robust, intrinsic, and discretizable gate framework independent of a fixed tensor factorization.

Abstract

In the standard circuit model, elementary gates are specified relative to a chosen tensor factorization and are therefore extrinsic to the ambient group . Writing , we introduce an intrinsic descriptor layer in by declaring as primitive the motions inside faithful embedded copies of , leading to the phase-free dictionary , and we also discuss the phase-inclusive variant. We show that decomposes into finitely many -homogeneous strata indexed by isotypic multiplicities, with stabilizers given by centralizers; the canonical two-level sector is organized by up to a gauge. Equipping 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 and hence . Full universality in follows by adjoining the abelian diagonal/global factors (equivalently, by passing to the two-level dictionary). Finally, we record a modular finite-alphabet interface by lifting Solovay--Kitaev approximation in through two-level embeddings.
Paper Structure (55 sections, 31 theorems, 82 equations)

This paper contains 55 sections, 31 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. Preliminary definitions and results
  4. Multi-qubit Hilbert space, ambient groups, and global-phase conventions
  5. Global phases and the choice of ambient group.
  6. Lie algebra, Hilbert--Schmidt geometry, and the exponential map
  7. Pauli matrices, Pauli strings, and normalization of generators
  8. Intrinsic locality via commutants and the canonical reference subclass
  9. Canonical one-qubit reference subclass.
  10. Elementary gates via Lie group embeddings
  11. Convention (two elementary dictionaries).
  12. Canonical local layer (reference subclass).
  13. Embeddings into U(N) and faithful unitary representations
  14. Logical Qubits and the Grassmannian Model
  15. Two-level subgroups and two-level embeddings
  16. ...and 40 more sections

Key Result

Proposition 2.5

Fix $N\in\mathbb{N}$ and let $G\in\{U(2),SU(2)\}$. There is a natural identification between Lie group embeddings $\phi\in \mathrm{Emb}(G,U(N))$ and faithful $N$-dimensional unitary representations of $G$. More precisely:

Theorems & Definitions (84)

  • Definition 2.1: $S$-local unitaries via commutants
  • Definition 2.2: Lie group embeddings
  • Remark 2.3: Why we stratify by $SU(2)$-embeddings
  • Definition 2.4: Unitary representation
  • Proposition 2.5: Embeddings $\boldsymbol{G\hookrightarrow U(N)}$ as faithful unitary representations
  • proof
  • Corollary 2.6: Conjugacy classes of embeddings and isotypic multiplicity data
  • proof
  • Definition 3.1: Two-level unitary and special-unitary subgroups
  • Remark 3.2: Non-canonical identifications
  • ...and 74 more