Synthesis and Upper Bound of Schmidt Rank of Bipartite Controlled-Unitary Gates

Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, it is shown 2(N-1)$2(N-1)$ generalized controlled-X (GCX) gates, 6 single-qubit rotations about the y- and z-axes, and N+5$N+5$ single-partite y- and z-rotation-types which are defined in this paper are sufficient to simulate a controlled-unitary gate Ucu(2 circle times N)$\mathcal {U}_{\text{cu}(2\otimes N)}$ with A$\text{A}$ controlling on C2 circle times CN$\mathbb {C}<^>2\otimes \mathbb {C}<^>N$. In the scenario of the unitary gate Ucd(M circle times N)$\mathcal {U}_{\text{cd}(M\otimes N)}$ with M >= 3$M\ge 3$ that is locally equivalent to a diagonal unitary on CM circle times CN$\mathbb {C}<^>M\otimes \mathbb {C}<^>N$, 2M(N-1)$2M(N-1)$ GCX gates and 2M(N-1)+10$2M(N-1)+10$ single-partite y- and z-rotation-types are required to simulate it. The quantum circuit for implementing Ucu(2 circle times N)$\mathcal {U}_{\text{cu}(2\otimes N)}$ and Ucd(M circle times N)$\mathcal {U}_{\text{cd}(M\otimes N)}$ are presented. Furthermore, it is found that Ucu(2 circle times 2)$\mathcal {U}_{\text{cu}(2\otimes 2)}$ with A$\text{A}$ controlling has Schmidt rank two, and in other cases the diagonalized form of the target unitaries can be expanded in terms of specific simple types of product unitary operators.

Automated summary

The authors propose a quantum circuit model for implementing complex quantum computation. They show that certain gates and rotations can simulate controlled-unitary gates and diagonal unitary gates. The quantum circuits for implementing the target gates are also presented.