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In papers, one of them being An Experimental Study of Shor's Factoring Algorithm on IBM Q is stated that replacing QFT with the semiclassical QFT (Kitaev's approach) reduces the needed number of qubits to solve a given problem with Shor's algorithm. Other than using a different approach - measuring one qubit of the period register each time contrary to the approach where multiple qubits are used is there a mathematical difference between sc-QFT and QFT?

More precisely, my question is: if QFT is mathematically described as

$$| x \rangle \rightarrow \frac{1}{\sqrt{N}} \sum_{y=0}^{N-1} \exp\left(\frac{2\pi ixy}{N}\right) | y\rangle,$$

how does the equation for sc-QFT look like?

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The semi-classical QFT only behaves the same as the QFT if you are immediately measuring the QFT's output. A mathematical description of the unitary effect of the QFT isn't going to work, because the semi-classical QFT isn't a unitary operation. It's a frequency space measurement operation.

You need to describe the semi-classical QFT in terms of the probability distribution of measurement results (as a function in the input state). You can also describe QFT+measure in this way, and confirm that their descriptions in terms of probability distribution are equal.

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