We concretely construct an extension of the controlled-U gate in qudit from some elementary gates. We also construct unitary transformation in two-qudit by means of the extended controlled-U gate and show the universality of it.
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 [S.Bullock, D.O'Leary, and G.K. Brennen, Phys. Rev. Lett. 94, 230502 (2005)] 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.
We construct an exchange gate with small elementary gates on the space of qudits, which consist of three controlled shift gates and three 'reverse' gates. This is a natural extension of the qubit case. We also consider a similar situation in Fock space, but in this case we find some differences. However, we can construct the exchange gate by making use of a generalized coherent operator based on the Lie algebra su (2), which is a well-known method in quantum optics. We also make a brief comment on 'imperfect clones'.
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.