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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?

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    $\begingroup$ The native gate set for IBM hardware is $CX, ID, RZ, SX, X$ $\endgroup$
    – KAJ226
    Apr 16 at 14:30
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    $\begingroup$ Note that $W = \frac{X+Y}{\sqrt{2}}$. $\endgroup$ Apr 16 at 17:59
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    $\begingroup$ No problem. It is the $\sqrt{X}$. qiskit.org/documentation/stubs/… $\endgroup$
    – KAJ226
    Apr 16 at 18:27
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    $\begingroup$ Native gate set used by IonQ comprised of single qubit rotations, rotations $Rx$, $Ry$, and $XX$ - Mølmer-Sørenson - two qubit gate. $\endgroup$ Apr 17 at 8:04
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    $\begingroup$ @MarkS: Thanks for suggestion, I edited the question. I plan after a few days to make a summary of comments in answer (of course with acknowledgement of the answerers). $\endgroup$ Apr 17 at 16:05
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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).

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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)
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