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I think similar to $R_z\big(\frac{\pi}{4}\big)$ gate named T gate, how to standardize the name $R_z\big(\frac{\pi}{8}\big)$?

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    $\begingroup$ There is no “official” name for this gate as there is for the T gate $\endgroup$ – rjh324 Apr 15 at 2:50
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    $\begingroup$ Just like number, only special one has name, like $\pi$ or $e$. It would be impossible to assign a name to each number... same goes with quantum gate. Only the special one has their own name, like Hadamard, T, S, CNOT, etc. The rest, we just say what they are... $\endgroup$ – KAJ226 Apr 15 at 4:50
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There is no widely recognizable one-letter name for the gate. However, $R\left(\frac{\pi}{8}\right)$ and the more precise $R_z\left(\frac{\pi}{8}\right)$ are clear and widely used.

In a slight abuse of notation, the gate is also occasionally called $\sqrt{T}$. This is justified by the fact that for any normal matrix $A$ with eigendecomposition $A=\sum_k\lambda_k|k\rangle\langle k|$ and any function $f$ analytic in a set containing all $\lambda_k$, we can compute $f(A)$ by applying $f$ to the eigenvalues $f(A) = \sum_k f(\lambda_k)|k\rangle\langle k|$ (see e.g. section 2.1.8 on page 75 in Nielsen & Chuang).

The reason $\sqrt{T}$ is an abuse of notation is that the square root in the complex domain is a multi-valued function. It is uniquely defined only for non-negative real numbers. Similarly, for matrices, the square root is uniquely defined only for positive semidefinite matrices, which $T$ is not. Consequently, the square root of $T$ is not unique and the notation relies on the expectation that the reader will guess that $\sqrt{T}$ is meant to denote

$$ \begin{pmatrix} 1 & 0 \\ 0 & e^{\pi i/8} \end{pmatrix} \equiv R_z\left(\frac{\pi}{8}\right) $$

and not

$$ \begin{pmatrix} 1 & 0 \\ 0 & e^{9\pi i/8} \end{pmatrix} \equiv R_z\left(\frac{9\pi}{8}\right) $$

despite the fact that the square of each matrix is $T$.

Note that this abuse of notation is used fairly consistently in quantum computing to refer to other gates, e.g. $\sqrt{\text{CNOT}}$, $\sqrt{\text{SWAP}}$, $\sqrt{X}$ and indeed $T$ which is occasionally denoted as $\sqrt{S}$.

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  • $\begingroup$ Yes,Look at the scenario where $T=\sqrt{S}$ is used, I choose to use $R_z(\frac{\pi}{8})$rather than random use$\sqrt{T}$. $\endgroup$ – Chuang Lee Apr 15 at 6:22
  • $\begingroup$ I don't understand. What do you mean? The answer basically says that you have three options: $R_z(\frac{\pi}{8})$ (the best), $R(\frac{\pi}{8})$ (a little less precise) and $\sqrt{T}$ (slight but common abuse of notation). $\endgroup$ – Adam Zalcman Apr 15 at 15:18

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