Optimal simulation of two-qubit Hamiltonians using general local operations

We consider the simulation of one bilocal Hamiltonian by another. Two otherwise isolated systems $A$ and $B$ undergo an evolution $e^{-iHt}$ where $H \neq H_A+H_B$ (not a sum of local terms). We consider the achievable space of Hamiltonians $H'$ such that the evolution $e^{-iH't}$ can be simulated by interspersing $e^{-iHt}$ with arbitrary local operations on $A$ and $B$. We find that any non-local Hamiltonian $sH'$ can be simulated by $H$ provided $s$ is small enough. For two-qubit Hamiltonians $H$, we find that the optimal protocol only requires local unitary operations. The optimal $s$ and the corresponding simulation protocol are obtained. Our solution has a simple geometric interpretation, and reveals a partial order structure in the set of two-qubit Hamiltonians.