Limitations on the simulation of non-sparse Hamiltonians

The problem of simulating sparse Hamiltonians on quantum computers is well-studied. The evolution of a sparse Hamiltonian H for time t can be simulated using O(‖Ht‖) operations, which is essentially optimal due to a no–fast-forwarding theorem. Here, we consider simulation of dense Hamiltonians. On the positive side, we show that some dense Hamiltonians can be simulated efficiently, such as those with graphs of small arboricity. However, we also show significant limitations on the simulation of non-sparse Hamiltonians. We generalize the no–fastforwarding theorem to dense Hamiltonians, ruling out generic simulations taking time o(‖Ht‖), even though ‖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 O(‖Ht‖), showing that known simulations based on discrete-time quantum walk cannot be dramatically improved in general.