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.