Synthesis and upper bound of Schmidt rank of the bipartite controlled-unitary gates

Abstract

Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, we show that $2(N-1)$ generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the $y$- and $z$-axes, and $N+5$ single-partite $y$- and $z$-rotation-types which are defined in this paper are sufficient to simulate a controlled-unitary gate ${\mathcal{U}}_{cu(2\otimes N)}$ with $A$ controlling on ${\mathbb{C}}^2\otimes {\mathbb{C}}^N$. In the scenario of the unitary gate ${\mathcal{U}}_{cd(M\otimes N)}$ with $M\ge 3$ that is locally equivalent to a diagonal unitary on ${\mathbb{C}}^M\otimes {\mathbb{C}}^N$, $2M(N-1)$ GCX gates and $2M(N-1)+10$ single-partite $y$- and $z$-rotation-types are required to simulate it. The quantum circuit for implementing ${\mathcal{U}}_{cu(2\otimes N)}$ and ${\mathcal{U}}_{cd(M\otimes N)}$ are presented. Furthermore, we find ${\mathcal{U}}_{cu(2\otimes 2)}$ with $A$ 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.

arXiv comment10 pages, 6 figures

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