Complexity Bounds for Hamiltonian Simulation in Unitary Representations

For any unitary representation $\rho$ on a finite-dimensional Hilbert space $V$ with differential $d\rho : \mathfrak{g} \to \mathfrak{u}(V)$ for the Lie algebra $\mathfrak g$, we consider the Hamiltonian evolution $$U_X(t) \coloneqq \rho(\exp(tX)) = e^{t\,d\rho(X)}, \qquad t\in\mathbb{R}.$$ For any complexification $X_\mathbb{C} = X_0 + \sum\limits_{\alpha\in\Delta} x_\alpha E_\alpha$ associated with the root system $\Delta$, we introduce the numerical invariants %\emph{root activity} and \emph{root curvature} functionals \begin{align*} \mathcal{A}_p(X) &\coloneqq \Bigl(\sum_{\alpha\in\Delta} |x_\alpha|^p \,\|d\rho(E_\alpha)\|_{\mathrm{op}}^p\Bigr)^{1/p}, \quad 1\le p<\infty\\ \mathcal{C}(X) &\coloneqq \Bigl(\sum_{\alpha\in\Delta} |\alpha(X_0)|^2\,|x_\alpha|^2 \,\|d\rho(E_\alpha)\|_{\mathrm{op}}^2\Bigr)^{1/2}, \end{align*} where $\|\cdot\|_{\mathrm{op}}$ is the operator norm on $\mathrm{End}(V)$. We first describe how the Hamiltonian $d\rho(X)$ is distributed along the directions of root spaces $\mathfrak{g}_\alpha$. Our main result shows that for each fixed $X\in\mathfrak{g}$ there exists a constant $C_X>0$ such that $$\bigl\| e^{t(d\rho(X_0)+d\rho(X_{\mathrm{root}}))} - e^{\frac{t}{2}d\rho(X_0)} e^{t d\rho(X_{\mathrm{root}})} e^{\frac{t}{2}d\rho(X_0)} \bigr\|_{\mathrm{op}} \le C_X\,t^{3}\,\bigl(\mathcal{C}(X)+\mathcal{A}_1(X_{\mathrm{root}})\bigr)$$ for all sufficiently small $|t|$. We also introduce a root-gate circuit model and test this on spin$-$chain Hamiltonians on $(\mathbb{C}^2)^{\otimes n}\subset\mathfrak{su}(2^n)$, 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.