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In Google's quantum computational supremacy experiment with their Sycamore transmon processor, they used single-qubit gates from $\{\sqrt{X},\sqrt{Y},\sqrt{W}\},$ with $W=\frac{X+Y}{\sqrt{2}}$.

Additionally for their two-qubit gates, they used something close to an $\mathsf{iSWAP}$ gate - something like a $\mathsf{SWAP}$ gate that adds a $i$ phase only to the $\vert11\rangle$ basis.

They say that supremacy experiments also like to use $\mathsf{CZ}$ gates, but one of the reasons they hint at these specific gates, in addition to being implementable on their devices, was that these gates appeared to maximize entanglement in a manner that made classical simulation more difficult.

(As an aside, classically we like to build most CMOS logic with $\mathsf{NAND}$ gates, although $\mathsf{NOR}$ gates also generate the set of Boolean functions. There are engineering reasons and also historical reasons why, as hinted at in this Quora question).