Robust quantum computation with d-level quantum systems (qudits) poses two

requirements: fast, parallel quantum gates and high fidelity two-qudit gates.

We first describe how to implement parallel single qudit operations. It is by

now well known that any single-qudit unitary can be decomposed into a sequence

of Givens rotations on two-dimensional subspaces of the qudit state space.

Using a coupling graph to represent physically allowed couplings between pairs

of qudit states, we then show that the logical depth of the parallel gate

sequence is equal to the height of an associated tree. The implementation of a

given unitary can then optimize the tradeoff between gate time and resources

used. These ideas are illustrated for qudits encoded in the ground hyperfine

states of the atomic alkalies $^{87}$Rb and $^{133}$Cs. Second, we provide a

protocol for implementing parallelized non-local two-qudit gates using the

assistance of entangled qubit pairs. Because the entangled qubits can be

prepared non-deterministically, this offers the possibility of high fidelity

two-qudit gates.