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Suppose I’m building a long circuit and I need to reuse a qubit from a previous step but I don’t want to use the reset operation (or I’m not using an IBM computer so I don’t even have the reset option).

Could I apply some Grover-style rotation of the qubit that I want to reset, like if I was searching for the state $|0\rangle$?

I know that the Grover algorithm presumes you are starting from a qubit in a uniform superimposition - but what would happen if I tried to do a Grover-like search for $|0\rangle$ starting from any arbitrary state instead of a uniform superimposition? Would I still get to $|0\rangle$ or would something weird happen?

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Unfortunately, as you say, something weird would happen. If you start with a qubit in an unknown quantum state, there are no unitary operations to map it to $|0\rangle$ because unitary operations must be reversible. Commonly, initialization in quantum computing is done by measuring the desired reset bit, and then applying $X$ to it if $|1\rangle$ is measured to get back to $|0\rangle$.

Grover's algorithm is unitary until the measurement step, so would be unable to help you reinitialize your qubit.

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  • $\begingroup$ Not sure I follow. If I start my qubit in an arbitrary state, why can’t I just use standard amplitude amplification to change its state all the way to my desired state. Granted I will need more than one iteration, but should still be possible, right? What would happen with the other qubits in the system if I apply a series of amplitude amplification steps to a single qubit? $\endgroup$
    – Fabio Dias
    Oct 26, 2022 at 8:27
  • $\begingroup$ Let's say you have a qubit in an arbitrary state $|\psi_1\rangle$ in one run of your algorithm and $|\psi_2\rangle$ in a second. Then any amplification process $A$ that re-initializes the qubit to $|0\rangle$ must have $A|\psi_1\rangle = A|\psi_2\rangle = |0\rangle$. Then such a process cannot be unitary (otherwise $A^\dagger|0\rangle = |\psi_1\rangle = |\psi_2\rangle$, a contradiction). This means any sort of Grover-like amplification process would be unable to do what you want since they're unitary. To reinitialize a qubit, you need to do something non-unitary like measurement. $\endgroup$
    – Chris E
    Oct 26, 2022 at 21:22
  • $\begingroup$ As for your second question, other than practical concerns that come from storing quantum information in current quantum computers, running an amplification process on a single qubit wouldn't affect the others $\endgroup$
    – Chris E
    Oct 26, 2022 at 21:24
  • $\begingroup$ For the first question, you actually used your own answer to prove your answer... I’m not trying to bring the qubit from an arbitrary state to an exact pure state of ket-zero. My point, which is not being fully understood maybe because I’m not being clear enough, is that you can use the amplitude amplification to rotate from some state to something “close enough” to a target state (this is what Grover does). The only issue is that you need to know the angle between your initial state and your target state so that you know how many times to rotate. $\endgroup$
    – Fabio Dias
    Oct 27, 2022 at 6:39
  • $\begingroup$ So my question was - is there any way to “guesstimate” the number of Grover iterations needed to rotate from some state to a target of ket-zero? My question isn’t really whether this is possible - this is certainly possible if you know how many iterations of the Grover algorithm to do. The question is “is there any clever way to find this number of iterations” or without any further information about the initial state are we hopeless? $\endgroup$
    – Fabio Dias
    Oct 27, 2022 at 6:42

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