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what is being said with "a single probe within $2^𝑛$ possible locations" It wants to say the process has translated the qubit (binary representation) into a single location in memory. What is the "probe"? And what is behind "degree of freedom"? A probe is an abstraction of a device that could visit a location (the degree of ...

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Ancilla-free solution: replace the two controlled-SWAPs in the "summary update" of Craig Gidney's solution with controlled-$Z$s between the second and fourth qubits in the diagram, and remove the third qubit. (That is, instead of swapping $|-\rangle$ with a $|+\rangle$ state stored in the second register, conditioned on $|q\rangle$ being set to 1, ...

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You're missing a bit of algebraic trickery. Remember that $\frac{1}{\sqrt{2}}=\sin(\pi/4)=\cos(\pi/4)$. Thus, $$\cos(\pi/8)/\sqrt{2}+\sin(\pi/8)/\sqrt{2}=\cos(\pi/8)\cos(\pi/4)+\sin(\pi/8)\sin(\pi/4)=\cos(\pi/4-\pi/8)=\cos(\pi/8)$$ by the double angle formula. Also, be careful of signs. It might be an amplitude is $\pm\sin(\pi/8)$, but when you take the ...

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