How efficiently can a 1-sparse Hamiltonian be simulated (quantum mechanically)?
Diagonal Hamiltonians are easy to simulate with 2 queries. It's very well explained (with a circuit diagram!) in Andrew Childs' PhD thesis on page 18, Rule 1.6. 

The main idea is to observe that if H is the Hamiltonian with diagonal entries d(j), then simulating H for time t maps the basis state $|j\rangle$ to $e^{-id(j)t}|j\rangle$. This is performed as follows: $|j, 0\rangle \rightarrow |j, d(j)\rangle$ (this uses 1 query), then $|j, d(j)\rangle \rightarrow e^{-id(j)t}|j, d(j)\rangle$ (this is a controlled phase flip on the first register controlled by the second), and then $e^{-id(j)t}|j, d(j)\rangle \rightarrow e^{-id(j)t}|j, 0\rangle$ (this uncomputes the answer using another query). 

Indeed, all 1-sparse Hamiltonians can be simulated with 2 queries. This is also explained in Andrew's thesis, towards the end of page 20. It might be a bit hard to understand since it's explained in the context of simulating a d-sparse Hamiltonian using edge coloring. But the essential idea is to reduce a d-sparse Hamiltonian to a sum of 1-sparse Hamiltonians and then simulate the 1-sparse Hamiltonian.