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

Abstract

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 [Multi-valued logic gates for quantum computation, Phys. Rev. A 62 (2000) 052309. [10]], Daboul et al. [Quantum gates on hybrid qudits, J. Phys. A Math. Gen. 36 (2003) 2525–2536. [5]], and Bartlett et al. [Quantum encodings in spin systems and harmonic oscillators, Phys. Rev. A 65 (2002) 052316. [17]]. 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. [Generalized Euler angle parameterization for SU(N), J. Phys. A Math. Gen. 35 (2002) 10467–10501. [15]] 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.