Basic gates sets

Question

There are several basic gate sets allowing to construct any gate on a quantum gate-based computer, e.g.:

• $H$, $T$, $CNOT$ (sometimes enriched to $H$, $T$, $S$, $X$, $CNOT$),
• rotations $Rx$, $Ry$ and $Rz$ and $CNOT,$
• Toffoli gate + $H,$
• Fredkin gate + $H.$

I am wondering whether there are any other universal sets usually used in quantum computation. What are advantages and drawbacks of these sets?

Answer 1

Community Wiki

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).

Answer 2

Here is a list of other basic gate sets based on comments to my question (I included a name of a comment author to brackets):

• The native gate set for IBM hardware is $CNOT$, $ID$, $Rz$, $X$ and $\sqrt{X}$ (by KAJ226)
• Google Sycamore gates: $\sqrt{X}$, $\sqrt{Y}$ and $\sqrt{W}$, where $W = (X + Y)/\sqrt{2}$ and gate similar to $iSWAP$ (described here and here) (by Mark S)
• Native gate set used by IonQ comprised of single qubit rotations $Rx$, $Ry$, and $XX$ which is Mølmer-Sørenson two qubit gate (by Egretta.Thula)