On page 3 here it is mentioned that:
However, building on prior works [32, 36, 38] recently it has been shown in [39] that to simulate $e^{−iHt}$ for an $s$-sparse Hamiltonian requires only $\mathcal{O}(s^2||Ht||\text{poly}(\log N, \log(1/\epsilon)))$, breaching the limitation of previous algorithms on the scaling in terms of $1/\epsilon$.
Questions:
What is meant by "simulate" in this context? Does it mean it takes $$\mathcal{O}(s^2||Ht||\text{poly}(\log N, \log(1/\epsilon)))$$ time to decompose $e^{-iHt}$ into elementary quantum gates given we know $H$. Or does it mean we can compute the matrix form of $e^{-iHt}$ in $$\mathcal{O}(s^2||Ht||\text{poly}(\log N,\log(1/\epsilon)))$$ time given we know the matrix form of $H$?
What does $||Ht||$ mean here? Determinant of $Ht$? Spectral norm of $Ht$? I checked the linked ppt and it seems to call $||H||$ the "norm" of $H$. Now I have no idea how they're defining "norm".