Add note about equation 10 numerical constants

docs/source/examples/shadows_tutorial.md

@@ -159,11 +159,15 @@ Based on the theorem, if the error rate of fidelity is $\epsilon$, i.e.
 |F(\rho,\sigma)-1|\leq\epsilon,
 \end{equation}
 then the minimum number of measurements $N$ (number of snapshots) should be:
-\begin{equation}
+```{math}
+:label: eq-label
 N = \frac{34}{\epsilon^2}\left\|\rho-\mathrm{Tr}(\rho)/{2^n}\mathbb{I}\right\|_{\mathrm{shadow}}^2
-\end{equation}
+```
 with the shadow norm upper bound of the random Pauli measurement $\left\|\cdot\right\|_{\mathrm{shadow}}\leq 2^k\|\cdot\|_\infty$ when the operator acting on $k$ qubits, we have $N\leq 34\epsilon^{-2}2^{2n}+\mathcal{O}(e^{-n})$. Based on Fuchs–van de Graaf inequalities and properties of $L_p$ norm, $\|\rho-\sigma\|_2\leq \|\rho-\sigma\|_1 \leq (1-F(\rho,\sigma))^{1/2}$, the $L_2$ norm distance between the state reconstructed through classical shadow estimation and the state prepared by the circuit is upperbound by the fidelity error rate $\epsilon$. The dependency of the bound number of measurements $N$ to achieve the error rate $\epsilon$ is depicted in function `n_measurements_tomography_bound`.
 
+```{note}
+Equation {eq}`eq-label` comes from equation S13 in the paper {cite}`huang2020predicting`. It contains some numerical constants and as noted by Remark 1 these constants result from a worst case argument. You may see values much smaller in practice. 
+```
 
 ```{code-cell} ipython3
 # error rate of state reconstruction epsilon < 1.