4
$\begingroup$

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?

$\endgroup$
9
  • 2
    $\begingroup$ The native gate set for IBM hardware is $CX, ID, RZ, SX, X$ $\endgroup$
    – KAJ226
    Apr 16, 2021 at 14:30
  • 1
    $\begingroup$ Note that $W = \frac{X+Y}{\sqrt{2}}$. $\endgroup$ Apr 16, 2021 at 17:59
  • 2
    $\begingroup$ No problem. It is the $\sqrt{X}$. qiskit.org/documentation/stubs/… $\endgroup$
    – KAJ226
    Apr 16, 2021 at 18:27
  • 1
    $\begingroup$ Native gate set used by IonQ comprised of single qubit rotations, rotations $Rx$, $Ry$, and $XX$ - Mølmer-Sørenson - two qubit gate. $\endgroup$ Apr 17, 2021 at 8:04
  • 2
    $\begingroup$ 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 $\endgroup$ Aug 8, 2022 at 11:05

2 Answers 2

6
$\begingroup$

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).

$\endgroup$
2
$\begingroup$

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)
$\endgroup$
3
  • 1
    $\begingroup$ 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 $|000\cdots 0\rangle$. $\endgroup$ Dec 29, 2021 at 2:13
  • $\begingroup$ @Mark S: See my original question where I mentioned this set. $\endgroup$ Dec 29, 2021 at 6:17
  • $\begingroup$ 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. $\endgroup$ Aug 9, 2022 at 3:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.