  What interactions are sufficient to simulate arbitrary quantum dynamics in a
composite quantum system? It has been shown that all two-body Hamiltonian
evolutions can be simulated using \emph{any} fixed two-body entangling
$n$-qubit Hamiltonian and fast local unitaries. By \emph{entangling} we mean
that every qubit is coupled to every other qubit, if not directly, then
indirectly via intermediate qubits. We extend this study to the case where
interactions may involve more than two qubits at a time. We find necessary and
sufficient conditions for an arbitrary $n$-qubit Hamiltonian to be
\emph{dynamically universal}, that is, able to simulate any other Hamiltonian
acting on $n$ qubits, possibly in an inefficient manner. We prove that an
entangling Hamiltonian is dynamically universal if and only if it contains at
least one coupling term involving an \emph{even} number of interacting qubits.
For \emph{odd} entangling Hamiltonians, i.e., Hamiltonians with couplings that
involve only an odd number of qubits, we prove that dynamic universality is
possible on an encoded set of $n-1$ logical qubits. We further prove that an
odd entangling Hamiltonian can simulate any other odd Hamiltonian and classify
the algebras that such Hamiltonians generate. Thus, our results show that up to
local unitary operations, there are only two fundamentally different types of
entangling Hamiltonian on $n$ qubits. We also demonstrate that, provided the
number of qubits directly coupled by the Hamiltonian is bounded above by a
constant, our techniques can be made efficient.
