Correction: Quantum Machine Intelligence (2022) 4:23

https://doi.org/10.1007/s42484-022-00079-9

The original online version of this article was revised: The authors noticed an error is in equation (7), proposition 2, page 3:

Incorrect Eq. --> \(\begin{array}{l}{\text{Pr}}\left[\underset{x\in \mathcal{M}}{{\text{sup}}}\left|{\widehat{f}}_{{\rho }_{{\text{train}}}}\left(x\right)-{\widehat{f}}_{\gamma }\left(x\right)\right|\ge \epsilon \right]\le \\\;\;\;\;\; {2}^{8}{\left(\frac{\sqrt{2d\gamma }{\text{diam}}\left(\mathcal{M}\right)}{3{M}_{\gamma }N\epsilon }\right)}^{2}{\text{exp}}\left(-\frac{D{\left(3{M}_{\gamma }N\epsilon \right)}^{2}}{4\left(d+2\right)}\right)\end{array}\)  

Correct Eq. --> \(\begin{array}{l}{\text{Pr}}\left[\underset{x\in \mathcal{M}}{{\text{sup}}}\left|{\widehat{f}}_{{\rho }_{{\text{train}}}}\left(x\right)-{\widehat{f}}_{\gamma }\left(x\right)\right|\ge \epsilon \right]\le \\\;\;\;\;\; {2}^{8}{\left(\frac{\sqrt{2d\gamma }{\text{diam}}\left(\mathcal{M}\right)}{3{M}_{\gamma }\epsilon }\right)}^{2}{\text{exp}}\left(-\frac{D{\left(3{M}_{\gamma }\epsilon \right)}^{2}}{4\left(d+2\right)}\right)\end{array}\)  

The proof of the proposition in the Appendix on pages 15 and 16 has also been corrected:

Incorrect Eq. --> \(\begin{array}{l}{\text{Pr}}\left[\underset{x\in \mathcal{M}}{{\text{sup}}}\left|{\widehat{f}}_{{\rho }_{{\text{train}}}}\left(x\right)-{\widehat{f}}_{\gamma }\left(x\right)\right|\ge \epsilon \right]\le \\\;\;\;\;\; {2}^{8}{\left(\frac{\sqrt{2d\gamma }{\text{diam}}\left(\mathcal{M}\right)}{3{M}_{\gamma }N\epsilon }\right)}^{2}{\text{exp}}\left(-\frac{D{\left(3{M}_{\gamma }N\epsilon \right)}^{2}}{4\left(d+2\right)}\right)\end{array}\)  

Correct Eq. ---> \(\begin{array}{l}{\text{Pr}}\left[\underset{x\in \mathcal{M}}{{\text{sup}}}\left|{\widehat{f}}_{{\rho }_{{\text{train}}}}\left(x\right)-{\widehat{f}}_{\gamma }\left(x\right)\right|\ge \epsilon \right]\le \\\;\;\;\;\; {2}^{8}{\left(\frac{\sqrt{2d\gamma }{\text{diam}}\left(\mathcal{M}\right)}{3{M}_{\gamma }\epsilon }\right)}^{2}{\text{exp}}\left(-\frac{D{\left(3{M}_{\gamma }\epsilon \right)}^{2}}{4\left(d+2\right)}\right)\end{array}\)  

Incorrect Eq. -->\({\text{Pr}}\left[\underset{x\in \mathcal{M}}{{\text{sup}}}\left|{\widehat{f}}_{\rho }\left(x\right)-{\widehat{f}}_{\gamma }\left(x\right)\right|\ge 3{M}_{\gamma }N\epsilon \right]\le B\)

Correct Eq. --->\({\text{Pr}}\left[\underset{x\in \mathcal{M}}{{\text{sup}}}\left|{\widehat{f}}_{\rho }\left(x\right)-{\widehat{f}}_{\gamma }\left(x\right)\right|\ge 3{M}_{\gamma }\epsilon \right]\le B\)

Incorrect Eq. -->\(\begin{array}{l}{\text{Pr}}\left[\underset{x\in \mathcal{M}}{{\text{sup}}}\left|{\widehat{f}}_{{\rho }_{{\text{train}}}}\left(x\right)-{\widehat{f}}_{\gamma }\left(x\right)\right|\ge \epsilon \right]\le \\\;\;\;\;\; {2}^{8}{\left(\frac{\sqrt{2d\gamma }{\text{diam}}\left(\mathcal{M}\right)}{3{M}_{\gamma }N\epsilon }\right)}^{2}{\text{exp}}\left(-\frac{D{\left(3{M}_{\gamma }N\epsilon \right)}^{2}}{4\left(d+2\right)}\right)\end{array}\)  

Correct Eq. --->\(\begin{array}{l}{\text{Pr}}\left[\underset{x\in \mathcal{M}}{{\text{sup}}}\left|{\widehat{f}}_{{\rho }_{{\text{train}}}}\left(x\right)-{\widehat{f}}_{\gamma }\left(x\right)\right|\ge \epsilon \right]\le \\\;\;\;\;\; {2}^{8}{\left(\frac{\sqrt{2d\gamma }{\text{diam}}\left(\mathcal{M}\right)}{3{M}_{\gamma }\epsilon }\right)}^{2}{\text{exp}}\left(-\frac{D{\left(3{M}_{\gamma }\epsilon \right)}^{2}}{4\left(d+2\right)}\right)\end{array}\)  

The original article has been corrected.