Reposted by Sayantan Sen
New preprint with @sayantansen.bsky.social and @marcotomamichel.bsky.social!!
In Fast and Sure-ious QPT, we describe a simple and rigorous way to lift any QST algorithm (in fidelity) to a QPT algorithm, while allowing for all guarantees to transfer!
scirate.com/arxiv/2609.3...
![Draw U1,...,Un independently and uniformly in [0,1], and compute the argmax of log(ai-log log(1/Ui). Call that Z.
Then Z is distributed proportionally to the weights a1,..,an](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:ac6qioenkpl3jmmnib3tpbhb/bafkreieentf6mvng3y5ge2rirgbjjd2p5zi4eucfpjj6bcam47pipufmpm)
![A short proof: here is the LaTeX code.
**Proof.** We have, for any $\color{blue}{\lambda} \in\mathbb{R}$,
\begin{align*}
\mathbb{E}[(X-\color{blue}{\lambda})^2]
&= \mathbb{E}[(X-\color{red}{\mathbb{E}[X]} + \color{red}{\mathbb{E}[X]} - \color{blue} {\lambda})^2] \\
&=\mathbb{E}[(X-\color{red}{\mathbb{E}[X]})^2 + 2(X-\color{red}{\mathbb{E}[X]})(\color{red}{\mathbb{E}[X]} - \color{blue} {\lambda}) + (\color{red}{\mathbb{E}[X]} - \color{blue} {\lambda})^2]\\
&=\underbrace{\mathbb{E}[(X-\color{red}{\mathbb{E}[X]})^2]}_{=\textrm{Var}[X]} + 2\underbrace{\mathbb{E}[X-\color{red}{\mathbb{E}[X]}]}_{=0}(\color{red}{\mathbb{E}[X]} - \color{blue} {\lambda})] + (\color{red}{\mathbb{E}[X]} - \color{blue} {\lambda})^2
\end{align*}
and that's all. (The first step is a trick known as *"hiding zero:"* writing $0=a-a$. 🤷)](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:ac6qioenkpl3jmmnib3tpbhb/bafkreieevv3petzx4kqxx3o6i4oqtugsjh52n7i2ellidxmetefboag6fi)
