Complexity Bounds for Hamiltonian Simulation in Unitary Representations

For any unitary representation ρ on a finite-dimensional Hilbert space V with differential dρ: mathfrak{g} → mathfrak{u}(V) for the Lie algebra mathfrak g, we consider the Hamiltonian evolution \[ U_X(t) \coloneqq ρexp(tX) = e^{t\,dρ(X)}, \qquad t\in\mathbb{R}. \] For any complexification X_mathbb{C} = X₀ + sumlimits_αinΔ x_αE_α associated with the root system Δ, we introduce the numerical invariants %root activity and root curvature functionals \begin{align*} \mathcal{A}_p(X) &\coloneqq \Biglsum_αinΔ |x_α|^p \|dρ(E_α\|_{\mathrm{op}}^p\Bigr)^{1/p}, \quad 1\le p<\infty\\ \mathcal{C}(X) &\coloneqq \Biglsum_αinΔ |α(X₀|^2\,|x_α|^2 \,\|dρE_α\|_{\mathrm{op}}^2\Bigr)^{1/2}, \end{align*} where \|cdot\|_op is the operator norm on End(V). We first describe how the Hamiltonian dρ(X) is distributed along the directions of root spaces mathfrak{g}_α. Our main result shows that for each fixed Xinmathfrak{g} there exists a constant C_X>0 such that \[ \bigl\| e^{tdρ(X₀+dρX_root)} - e^{\frac{t}{2}dρX₀} e^{t dρX_root} e^{\frac{t}{2}dρX₀} \bigr\|_{\mathrm{op}} \le C_X\,t^{3}\,\biglmathcal{C}(X+\mathcal{A}_1X_root\bigr) \] for all sufficiently small |t|. We also introduce a root-gate circuit model and test this on spin-chain Hamiltonians on \(mathbb{C}²^{\otimes n}\subset\mathfrak{su}2ⁿ\), where root spaces are spanned by matrix units, mathcal{A}_p, and mathcal{C}, which gives sharper complexity bounds and dimension-free representation-theoretic invariants.