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Within the theory of quantum gates, a common pair of single-qubit phase gates are the $P(\pi/2)=S$ and $P(\pi/4)=T$ gates, with

$$S= \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix},\:T = \begin{bmatrix} 1 & 0 \\ 0 & e^{i \frac{\pi}{4}} \end{bmatrix}.$$

We have, for example, $T^4=S^2=Z$.

See this Wikipedia article.

Within the theory of presentations of $\mathrm{PSL}(2, \mathbb Z)$ (the modular group), we have the two generators, $S$ and $T$, with:

$$S : z\mapsto -\frac1z,\:T : z\mapsto z+1.$$

We have, for example, $S^2=(ST)^3=I$

See this Wikipedia article.

But, in matrix form, these generators do not look like those above:

$$S = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, \: T = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}.$$

Are the $S$ and $T$ labels used in quantum computation just coincidentally the same as those used to describe generators of the modular group? Or is there some deeper relation that I'm not immediately seeing? What is the origin of $S$ and $T$ gates used in quantum computation?

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    $\begingroup$ Thanks... I can agree that there is a relation and the folks working on these back in the day certainly might have had something in mind, but my knowledge of group theory is not strong enough to follow. Just naively, as matrices $S$ in group-theory land looks like $-iY$ in quantum-computing land, while $T$ in group-theory land is not even unitary. Plus they obey different relations (e.g. we have in group-theory land that $S^2=(ST)^3=I$, but this is not so with quantum gates AFAICT.) $\endgroup$
    – Mark Spinelli
    Commented Jun 14, 2021 at 14:01
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    $\begingroup$ I think this is merely a coincidence of notation. I think that the more interesting aspects of $S$, $T$, (and if you also throw in $H$) is that they generate a so-called $\mathcal{S}$-arithmetic subgroup of the projective unitary group. The astonishing property of this such subgroups is not only that they generate a topologically dense subgroup (which enables universal quantum computation) but that short approximating words exist which means that you can always approximate (up to $\epsilon$) an element of $PU(2)$ with a word of length $O(\log(1/\epsilon)$ in the generators $H, S,$ and $T$. $\endgroup$
    – Condo
    Commented Jun 14, 2021 at 15:12
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    $\begingroup$ You should only look at the group relations not about unitarity in the modular group's defining representation. The more relevant representation for the modular group is the one that comes secondarily through putting a Conformal Field Theory on a torus. $\endgroup$
    – AHusain
    Commented Jun 14, 2021 at 18:12
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    $\begingroup$ @MarkS it is in fact a strengthening of Solovay-Kitaev because the exponent $c$ in $O(\log^c(1/\epsilon))$ (from Solovay-Kitaev) is equal to $1$ for such gate sets. $\endgroup$
    – Condo
    Commented Jun 16, 2021 at 13:53
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    $\begingroup$ For more information, I would recommend reading arxiv.org/pdf/1704.02106.pdf or this hand written letter by Peter Sarnak to Scott Aaronson and Andy Pollington publications.ias.edu/sites/default/files/… $\endgroup$
    – Condo
    Commented Jun 16, 2021 at 13:56

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I believe Neilsen and Chuang were the first to use this particular notation. Previous work had referred to $S$ and $T$ as $\sigma_z^{1/2}$ and $\sigma_z^{1/4}$, respectively (Boykin et al. 1999). The use of $S$ may have been inspired by Deutsch's "S-matrix" (Deutsch 1989), though this was really a root-of-NOT gate. The use of $T$ may have been inspired by the transformation "T" matrix of a universal beam splitter (DiVincenzo 1989), which is equivalent to the modern $T$ matrix, up to a global phase, for certain parameter values.

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