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.

10 pages, 6 figures

References in corpus (12)

• Fast universal quantum control above the fault-tolerance threshold in silicon
• Low-cost Fredkin gate with auxiliary space
• Optimal synthesis of the Fredkin gate in a multilevel system
• Universal quantum multi-qubit entangling gates with auxiliary spaces
• Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates
• All unitaries having operator Schmidt rank 2 are controlled unitaries
• On the Schmidt-rank-three bipartite and multipartite unitary operator
• Efficient Fusion of Photonic W-states with Nonunitary Partial-swap Gates
• Blind quantum computation with a heralded single photon source
• Two-party LOCC convertibility of quadpartite states and Kraus-Cirac number of two-qubit unitaries
• Constructing three-qubit unitary gates in terms of Schmidt rank and CNOT gates
• Collective unitary evolution with linear optics by Cartan decomposition