Unitary operations are the building blocks of quantum programs. Our task is to design efficient 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.
We present explicit descriptions for the swap gate and the iswap gate in any arbitrary dimension d≥2, in terms of permutation matrices. Moreover, we unify these gates by introducing a more general gate xSWAP which includes SWAP and iSWAP for x=1 and x=i (i.e. −1), respectively. The higher dimensional xSWAP e.g., the swap and iswap gates for d>2 serve as quantum logic gates that operate on two d-level qudits. For d=2, it is well known that iSWAP unlike SWAP is universal for quantum computing. We will prove this fact for xSWAP in any dimension d, when x=±1. Our explicit representation of xSWAP by a permutation matrix facilitates the proof, greatly.