  Unitary operations are the building blocks of quantum programs. Our task is
to design effcient or optimal implementations of these unitary operations by
employing the intrinsic physical resources of a given n-qubit system. The most
common versions of this task are known as Hamiltonian simulation and gate
simulation, where Hamiltonian simulation can be seen as an infinitesimal
version of the general task of gate simulation. We present a Lie-theoretic
approach to Hamiltonian simulation and gate simulation. From this, we derive
lower bounds on the time complexity in the n-qubit case, generalizing known
results to both even and odd n. To achieve this we develop a generalization of
the so-called magic basis for two-qubits. As a corollary, we note a connection
to entanglement measures of concurrence-type.
