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I'm studying Shor's algorithm. but I see from the beginning this empty circle. what this circle means??

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It is a "negative" control.

In other terms, the gate in your circuit will apply a NOT on the last qubit if and only if the following conditions are both met:

  • $\vert x_3\rangle = \vert 0 \rangle$
  • the first ancilla qubit is in the state $\vert 1 \rangle$.

A "normal" control (i.e. not an empty circle) would change the first condition to $\vert x_3 \rangle = \vert 1 \rangle$.

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  • $\begingroup$ Thank you very much!! you are so kind :) Have a nice day!! $\endgroup$ – 유도경 Feb 9 at 7:33

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