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 \[ \mathcal{R}(d_{\mathrm{eff}}, M) \;\geq\; \Omega\!\left(\sqrt{\frac{d_{\mathrm{eff}}}{M}}\right), \] where $d_{\mathrm{eff}} = \dim(\mathcal{O}_O)$ is the real dimension of the observable's adjoint orbit under the dynamical Lie group. For balanced Pauli observables $O = Z^{\otimes n}$, quotienting the dynamical Lie algebra by the stabilizer subgroup yields the complex Grassmannian $Gr(2^{n-1}, 2^n)$ of real dimension $d_{\mathrm{eff}} = 2^{2n-1} = \frac{1}{2}4^n \approx \frac{1}{2}\dim(\mathfrak{g})$, which matches upper bounds of Caro et al.~(2022) and Ohno~(2025) up to a universal constant and resolves the exact tightness rate $\Theta(\sqrt{\dim(\mathfrak{g})/M})$. For rank-1 projective observables, the orbit contracts instead to complex projective space $\mathbb{CP}^{2^n-1}$ with dimension $d_{\mathrm{eff}} = 2(2^n-1)$. Sample requirements therefore explode exponentially with qubit count. The scaling identifies Wang \& Wu's (2025) dimension-independent $\mathcal{O}(1/\sqrt{M})$ 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 $SU(2^n)$ simultaneously suppresses gradient variances as $\sim 1/\dim(\mathfrak{g})$ and drives sample complexity up to $\sim \sqrt{\dim(\mathfrak{g})/M}$.