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Background: There is a single qubit quantum gate of the form $$ \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -i \\ 1 & i \end{pmatrix}. $$ As far as I know, this was originally introduced as the $T$ gate in Gottesman's thesis https://arxiv.org/abs/quant-ph/9705052 (see equation A.5). In https://arxiv.org/abs/quant-ph/0403025 the authors call a similar version of this gate the $T$ gate.

However, as we know, this eventually fell out of favor since the $T$ gate almost universally means the $\pi/8$ rotation gate (i.e., $T = S^{1/2} = Z^{1/4}$).

Question: Since $T$ is off limits, what should we call this gate? Or does this gate already have a very standard name that isn't $T$?

Things to consider:

  • In https://arxiv.org/abs/1702.06990 the authors call this the $M_3$ gate (they seem to claim Gottesman calls it that in his thesis but I am not sure I can find where he calls it anything other than $T$?).

  • In our recent paper https://arxiv.org/abs/2305.07023 we call it the $M$ gate but admittedly I am asking this question specifically because I think that is a poor name (for starters it literally doesn't stand for anything related to any of its properties).

  • In https://threeplusone.com/pubs/on_gates.pdf the author cites Craig Gidney's stim python package and calls this (and the ones like it) the "axis-cycling gates". The name used is $C$ but I personally am not a fan since this usually denotes the Clifford group or Clifford hierarchy (or even the two qubit canonical gates).

  • There is a twitter post https://twitter.com/CraigGidney/status/1393069730785857536 where some other choices are listed. One choice therein seems to be: ``face" gates denoted by $F$ since this gate always maps the faces of a octahedron to another face.

My thoughts: I personally think ``axis cycling gates" above is pretty descriptive since there are $16$ of these in total and each is defined by their conjugation action: cycling the axes up to a factor of $-1$. The gate above is sort of the canonical version as it acts as $X \to Y$, $Y \to Z$, and $Z \to X$. Maybe $A$ would be a good name?

Another property we point out in our above paper is this gate's importance to the binary tetrahedral group $2T$. In particular, $2T$ contains the the $8$ element Pauli group (from $SU(2)$) and the 16 axis cycling gates and that's it. In other words, the (special) Paulis plus the above gate generate $2T$. However, the obvious choice of name $T$ is out but maybe there is some other letter/notation related to this property.

Edit (7/26/23) In the 3rd version of our paper, we ended up going with $F$ for this gate and calling it the "facet" gate. This was inspired by what Mark said below, however, "face" and "phase" are near-homophones and so we thought facet made more sense (besides it is technically more accurate). This coincides with a comment made by Markus Heinrich in this question. Compare with figure 3 here.

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  • $\begingroup$ "What should we call this gate" sounds pretty opinion-based. A better question would be whether there are standard names for this gate. $\endgroup$ May 22, 2023 at 21:39
  • $\begingroup$ @MarkS Fixed. Sorry about that. $\endgroup$ May 23, 2023 at 0:02
  • $\begingroup$ @NorbertSchuch I reorganized to make it clear I am asking both questions. I think it's important to hear preferences but I agree that the primary aim of the question should be to find out if there is some notation already in wide use. $\endgroup$ May 23, 2023 at 0:16
  • $\begingroup$ ... also, the aim of StackExchange is not to collect opinions. -- On a more opinion-based note, I don't think everything needs its own name/letter, and conversely, it happens all the time that several things have the same name. $\endgroup$ May 23, 2023 at 7:56
  • $\begingroup$ F^24 gives the Identity matrix, maybe this could help $\endgroup$ Aug 6, 2023 at 14:58

2 Answers 2

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I advocate for calling your $M$ gate by the letter $F$ for face, for the reasons depicted in Ciaran's picture.

One slight problem with $A$ is that $|A\rangle$ usually means (for surface code people at least) the magic state that when injected gives the $T$ gate (this usage was in place for the Bravyi-Kitaev Magic state paper if not before). Wherever possible I think it's nice to call the magic state and the gate it gives by the same name.

I have to admit that when I tried to use $F$ for the face gate, I was prevented by the fact that we were already using $F$ for Fidelity (see eq 104 of https://arxiv.org/abs/1808.00128), but I don't think those two concepts clash too regularly.

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In stim, this gate is called C_XYZ and its inverse is called C_ZYX. This was the result of that twitter poll you referenced. The 'C' sort of collides with 'Clifford' and also with 'controlled', which is not ideal.

The 'C_XYZ' Gate

Right handed period 3 axis cycling gate, sending X -> Y -> Z -> X.

Stabilizer Generators:

X -> +Y
Z -> +X

Bloch Rotation:

Axis: +X+Y+Z
Angle: 120 degrees

Unitary Matrix:

[+1-i, -1-i]
[+1-i, +1+i] / 2

Decomposition (into H, S, CX, M, R):

# The following circuit is equivalent (up to global phase) to `C_XYZ 0`
S 0
S 0
S 0
H 0
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