Nested Bloch-Sphere Rotations Achieve Arbitrarily Accurate SU(2) Gates — E8 Intelligence Research

FINDING: Composite pulse sequences on the Bloch sphere achieve arbitrarily accurate SU(2) rotations by nesting error-compensating rotations, with the key structural insight being that any single-qubit gate is a 3D rotation in SU(2) ≅ SO(3), parameterized by Euler angles on the sphere. MATH: - Qubit state: |ψ⟩ = cos(θ/2)|0⟩ + eiφ sin(θ/2)|1⟩, with θ ∈ [0,π], φ ∈ [0,2π) — the Bloch sphere coordinates. - SU(2) generators: Pauli matrices σ_x, σ_y, σ_z, satisfying [σ_i, σ_j] = 2iεᵢⱼₖσ_k. - Rotation operator: R_n̂(θ) = exp(−iθ n̂·σ/2) = cos(θ/2)I − i sin(θ/2)(n̂·σ). - Composite pulse: U_total = Π_k Rn̂_k(θ_k), designed so that first-order (and higher) error terms in pulse amplitude/duration vanish: ∂U_total/∂ε |ε=0 = 0, up to desired order N. - The arXiv paper (1810.03833) explicitly constructs sequences with arbitrary N, using nested commutators of rotation axes to cancel error terms — a Lie-algebraic approach. CONNECTION: - The Bloch sphere is the unit sphere S², and SU(2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com