Synthesis of multi-qudit Hybrid and d-valued Quantum Logic Circuits by   Decomposition
  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. 
Faisal Shah Khan and Marek Perkowski