# Basic gates sets

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?

• The native gate set for IBM hardware is $CX, ID, RZ, SX, X$ Apr 16, 2021 at 14:30
• Note that $W = \frac{X+Y}{\sqrt{2}}$. Apr 16, 2021 at 17:59
• No problem. It is the $\sqrt{X}$. qiskit.org/documentation/stubs/… Apr 16, 2021 at 18:27
• Native gate set used by IonQ comprised of single qubit rotations, rotations $Rx$, $Ry$, and $XX$ - Mølmer-Sørenson - two qubit gate. Apr 17, 2021 at 8:04
• Criteria for universality of quantum gates can be found inhttps://journals.aps.org/pra/abstract/10.1103/PhysRevA.105.052602 and arxiv version arxiv.org/abs/2111.03862 see also arxiv.org/abs/1610.00547 and arxiv.org/abs/1609.05780 Aug 8, 2022 at 11:05

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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).

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)
• I’d also add H/Toffoli if only for some nice theoretical reasons- namely, that set never introduces a complex phase if acting only on $|000\cdots 0\rangle$. Dec 29, 2021 at 2:13
• @Mark S: See my original question where I mentioned this set. Dec 29, 2021 at 6:17
• Note that $\sqrt{X}$, $\sqrt{W}$ and the fSim gate (i.e. the iSWAP-like gate) are sufficient for universality, i.e. we don't actually need $\sqrt{Y}$. See VII F 2 on page $30$ in the supplement to the QS paper. Aug 9, 2022 at 3:47