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I want to implement a unitary $U\,,$ $$U=\text{exp}(-it|u\rangle\langle u|)\,,$$ where $|u\rangle$ is a known state.

Are there any methods to do this efficiently?

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  • $\begingroup$ What do you know about $|u\rangle$? For example, do you know that there exists a unitary $V$ that you can construct such that $|u\rangle=V|0\rangle$. $\endgroup$
    – DaftWullie
    Apr 12 at 6:30
  • $\begingroup$ yeah all information of |u> is known, suppose there exists a unitary V one can construct such that |u⟩=V|0⟩ $\endgroup$
    – mingo
    Apr 12 at 9:19

1 Answer 1

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Let's assume that we know the circuit $V$ the constructs $|u\rangle$, i.e. $$ V|0\rangle=|u\rangle. $$ So, this reduces our problem to creating the evolution $$ V^\dagger UV=e^{-it|0\rangle\langle 0|}. $$ This is quite straightforward. For our computational register, we can ask "is it in the state $|0\rangle$?" essentially by applying $X$ on every qubit, and doing a multi-controlled-not targeting an ancilla initially in $|0\rangle$. Next, you simply apply a phase gate $|0\rangle\langle 0|+e^{-it}|1\rangle\langle 1|$ on the ancilla, before undoing the entanglement between ancilla and computational register by repeating the multi-controlled-not.

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  • $\begingroup$ Thank you very much! Your method is quite useful. $\endgroup$
    – mingo
    Apr 13 at 9:24

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