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We consider the simulation of the dynamics of one nonlocal Hamiltonian byanother, allowing arbitrary local resources but no entanglement nor classicalcommunication. We characterize notions of simulation, and proceed to focus ondeterministic simulation involving one copy of the system. More specifically,two otherwise isolated systems $A$ and $B$ interact by a nonlocal Hamiltonian$H \neq H_A+H_B$. We consider the achievable space of Hamiltonians $H'$ suchthat the evolution $e^{-iH't}$ can be simulated by the interaction $H$interspersed with local operations. For any dimensions of $A$ and $B$, and anynonlocal Hamiltonians $H$ and $H'$, there exists a scale factor $s$ such thatfor all times $t$ the evolution $e^{-iH'st}$ can be simulated by $H$ acting fortime $t$ interspersed with local operations. For 2-qubit Hamiltonians $H$ and$H'$, we calculate the optimal $s$ and give protocols achieving it. The optimalprotocols do not require local ancillas, and can be understood geometrically interms of a polyhedron defined by a partial order on the set of 2-qubitHamiltonians.Comment: (1) References to related work, (2) protocol to simulate one  two-qudit Hamiltonian with another, and (3) other related results added. Some  proofs are simplifie

Optimal simulation of two-qubit Hamiltonians using general local operations