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

• URL: http://arxiv.org/abs/2209.04799v1
• Date: Sun, 11 Sep 2022 06:24:24 GMT
• Title: Synthesis and upper bound of Schmidt rank of the bipartite controlled-unitary gates
• Authors: Gui-Long Jiang, Hai-Rui Wei, Guo-Zhu Song, and Ming Hua
• Abstract summary: 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 are required to simulate it. The quantum circuit for implementing $mathcalU_cu(2otimes N)$ and $mathcalU_cd(Motimes N)$ are presented.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $M\geq3$ 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 $\mathcal{U}_{cu(2\otimes2)}$ 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.

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