Who was the first to call the phase gates $P(\pi/2)$ and $P(\pi/4)$ the $S$ and $T$ gates, and were they motivated by generators of the modular group?

Question

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, $T^4=S^2=Z$.

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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, $S^2=(ST)^3=I$. But, in matrix form, these generators do not look like those above:

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

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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?

Answer 1

I believe Nielsen 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.