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

TL;DR

This paper develops a method to synthesize controlled-unitary gates in quantum circuits using a minimal number of generalized controlled-X gates and single-qubit rotations, providing upper bounds on their Schmidt rank.

Contribution

The paper introduces a new synthesis approach for controlled-unitary gates based on Cartan decomposition, establishing upper bounds on the number of gates needed and analyzing Schmidt rank.

Findings

$2(N-1)$ GCX gates suffice for controlled-unitary gates on $2 imes N$ systems.

$2M(N-1)$ GCX gates are needed for certain diagonal unitaries on $M imes N$ systems.

Controlled-unitary gates with $A$ controlling on $bC^2 imes bC^N$ have Schmidt rank two.

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.