  This paper concerns the efficient implementation of quantum circuits for
qudits. We show that controlled two-qudit gates can be implemented without
ancillas and prove that the gate library containing arbitrary local unitaries
and one two-qudit gate, CINC, is exact-universal. A recent paper (PRL 94
230502) describes quantum circuits for qudits which require O(d^n) two-qudit
gates for state synthesis and O(d^{2n}) two-qudit gates for unitary synthesis,
matching the respective lower bound complexities. In this work, we present the
state synthesis circuit in much greater detail and prove that it is correct.
Also, the (n-2)/(d-2) ancillas required in the original algorithm may be
removed without changing the asymptotics. Further, we present a new algorithm
for unitary synthesis, inspired by the QR matrix decomposition, which is also
asymptotically optimal.

Quantum Physics