  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.
