Synthesis of multi-qudit hybrid and d-valued quantum logic circuits by decomposition

Recent research in generalizing quantum computation from 2-valued qudits to dvalued 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 dvalued 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 parametrization. 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.