Abstract
The problem of simulating sparse Hamiltonians on quantum computers is well studied. The evolution of a sparse $N \times N$ Hamiltonian $H$ for time $t$ can be simulated using $\O(\norm{Ht} \poly(\log N))$ operations, which is essentially optimal due to a no--fast-forwarding theorem. Here, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, ruling out generic simulations taking time $\o(\norm{Ht})$, even though $\norm{H}$ is not a unique measure of the size of a dense Hamiltonian $H$. We also present a stronger limitation ruling out the possibility of generic simulations taking time $\poly(\norm{Ht},\log N)$, showing that known simulations based on discrete-time quantum walk cannot be dramatically improved in general. On the positive side, we show that some non-sparse Hamiltonians can be simulated efficiently, such as those with graphs of small arboricity.
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