I've been reading this paper, but don't understand what their optimal method really is, and how it can be realized as a quantum circuit.

The paper mentions the "Schur transform" which has a circuit provided in this paper. But is this Schur transform the only thing taking place?

The paper also gives this diagram, but it isn't very thoroughly explained. What is this projector?

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Does anybody know this method, and if so, is there a simpler explanation leaning more toward quantum circuits? How is the Schur transform related to estimating inner products between quantum states?

Also, how is it that the algorithm's complexity doesn't depend on the dimension of the two states as a whole; it only depends on the number of qubits (or qudits, and $d$) in the state.



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