Replication Package and Codebase: A Tight Lower Bound on Generalization in Quantum Machine Learning via Dynamical Lie Algebra Dimension

Orbit dimension, not parameter count, bounds quantum sample complexity. For continuous parameterized quantum circuits on n qubits, the minimax estimation risk obeys \[ R(deff, M) \;≥\; Ω\!(√{{deff}{M}}), \] where deff = (OO) is the real dimension of the observable's adjoint orbit under the dynamical Lie group. For balanced Pauli observables O = Z⊗ n, quotienting the dynamical Lie algebra by the stabilizer subgroup yields the complex Grassmannian Gr(2ⁿ⁻¹, 2ⁿ) of real dimension deff = 2²ⁿ⁻¹ = 1/24ⁿ ≈ 1/2(g), which matches upper bounds of Caro et al.~(2022) and Ohno~(2025) up to a universal constant and resolves the exact tightness rate Θ(√(g)/M). For rank-1 projective observables, the orbit contracts instead to complex projective space CP2ⁿ-1 with dimension deff = 2(2ⁿ-1). Sample requirements therefore explode exponentially with qubit count. The scaling identifies Wang & Wu's (2025) dimension-independent O(1/√M) uniform convergence bound as an artifact of uniform stability rather than true estimation power. PennyLane benchmarks across $n=1,2,3,4$ qubits confirm this exponential threshold. Barren plateaus and statistical sample bottlenecks stem from an identical geometric origin: volume growth in SU(2ⁿ) simultaneously suppresses gradient variances as ~ 1/(g) and drives sample complexity up to ~ √(g)/M.