  Recent research in generalizing quantum computation from 2-valued qudits to
d-valued qudits has shown practical advantages for scaling up a quantum
computer. A further generalization leads to quantum computing with hybrid
qudits where two or more qudits have different finite dimensions. Advantages of
hybrid and d-valued gates (circuits) and their physical realizations have been
studied in detail by Muthukrishnan and Stroud (Physical Review A, 052309,
2000), Daboul et al. (J. Phys. A: Math. Gen. 36 2525-2536, 2003), and Bartlett
et al (Physical Review A, Vol.65, 052316, 2002). In both cases, a quantum
computation is performed when a unitary evolution operator, acting as a quantum
logic gate, transforms the state of qudits in a quantum system. Unitary
operators can be represented by square unitary matrices. If the system consists
of a single qudit, then Tilma et al (J.Phys. A: Math. Gen. 35 (2002)
10467-10501) have shown that the unitary evolution matrix (gate) can be
synthesized in terms of its Euler angle parameterization. However, if the
quantum system consists of multiple qudits, then a gate may be synthesized by
matrix decomposition techniques such as QR factorization and the Cosine-sine
Decomposition (CSD). In this article, we present a CSD based synthesis method
for n qudit hybrid quantum gates, and as a consequence, derive a CSD based
synthesis method for n qudit gates where all the qudits have the same
dimension.
