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My question concerns this algorithm: Fast Quantum Algorithm for Numerical Gradient Estimation by Stephen P Jordan https://arxiv.org/pdf/quant-ph/0405146

I am unable to understand this step of the algorithm (bottom right corner of page 1, some of the explanation continues onto page 2). enter image description here

I would expect the phases to be $e^{i 2\pi f(\delta)}$ based on what is described. It seems a lot of steps are skipped so I was hoping someone more familiar with the algorithm could reconstruct it.

The use of $N$ is to convert between the actual real numbers in the calculation and the fixed point integer encodings of them. I don't understand why $l$ multiplies $\delta$ , based on what appears before in the paper it seems that $l/2$ is small displacement in the input used for the estimation of the gradient so $f(\delta/N - l\vec{1}/2)$ makes sense but in the paper $l$ is also multiplied to $\delta$.

The $N$ mutiplied outside the function application also swaps between real number and fixed point encoding, the division by $l$ is from the denominator of the gradient estimation (though I am not sure why it is $l$ and not $l/2$. I am not sure what the $m$ does.

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