Universal Barenco quantum gates via a tunable non-collinear interaction

The Barenco gate~($\mathbb{B}$) is a type of two-qubit quantum gate based on which alone universal quantum computation can be achieved. Each $\mathbb{B}$ is characterized by three angles ($\alpha,\theta$, and $\phi$) though it works in a two-qubit Hilbert space. Here we design $\mathbb{B}$ via a non-collinear interaction $V|r_1r_2\rangle\langle r_1r_3|+$H.c., where $|r_i\rangle$ is a state that can be excited from a qubit state and $V$ is adjustable. We present two protocols of $\mathbb{B}$. The first~(second) protocol consists of two~(six) pulses and one~(two) wait period(s), where the former causes rotations between the qubit states and excited states, and the latter induces gate transformation via the non-collinear interaction. In the first protocol, the variable $\phi$ can be tuned by varying phases of external controls, and the other two variables $\alpha$ and $\theta$, tunable via adjusting the wait duration, have a linear dependence upon each other. Meanwhile, the first protocol can give rise to the CNOT and Controlled-Y gates. In the second protocol, $\alpha,\theta$, and $\phi$ can be varied by changing the interaction amplitudes and wait durations, and the latter two are dependent on $\alpha$ non-linearly. Both protocols can also lead to another universal gate when $\{\alpha,\phi\}=\{1/4,1/2\}\pi$ with appropriate parameters. Implementation of these universal gates is analyzed based on the van der Waals interaction of neutral Rydberg atoms.