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 $$U_X(t)\rho (\exp (tX))=e^{td\rho (X)},t\in \mathbb{R}\mathrm{.}$$ For any complexification $X_{\mathbb{C}}=X_0+\sum _{\alpha \in \mathrm{\Delta }}{x}_{\alpha }{E}_{\alpha }$ associated with the root system $\mathrm{\Delta }$, we introduce the numerical invariants %\emph{root activity} and \emph{root curvature} functionals \begin{align*} $\mathcal{A}$p(X) &\coloneqq \Bigl(\sum{α\inΔ} |x_α|^p ,|dρ(E_α)|{$\mathrm{o}\mathrm{p}$}^p\Bigr)^{1/p}, \quad 1\le p<\infty\ $\mathcal{C}$(X) &\coloneqq \Bigl(\sum{α\inΔ} |α(X_0)|^2,|x_α|^2 ,|dρ(E_α)|_{$\mathrm{o}\mathrm{p}$}^2\Bigr)^{1/2}, \end{align*} where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ is the operator norm on $\mathrm{E}\mathrm{n}\mathrm{d}(V)$. 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 $$\parallel e^{t(d\rho (X_0)+d\rho (X_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}))}-e^{\frac{t}{2}d\rho (X_0)}{e}^{td\rho (X_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}})}{e}^{\frac{t}{2}d\rho (X_0)}{\parallel }_{\mathrm{o}\mathrm{p}}\le C_X{t}^3(\mathcal{C}(X)+{\mathcal{A}}_1(X_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}))$$ for all sufficiently small $\mathrm{\mid }t\mathrm{\mid }$. We also introduce a root-gate circuit model and test this on spin$-$chain Hamiltonians on $({\mathbb{C}}^2{)}^{\otimes n}\subset \mathfrak{s}\mathfrak{u}(2^n)$, 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.