# How to determine the basis state with maximum amplitude without measurement?

Suppose I have two quantum registers described respectively by the quantum states $$| \psi_1 \rangle = \sum_i \alpha_i |i \rangle$$ and $$|\psi_2 \rangle = |0\rangle$$. I would like to implement a CNOT gate where the target is $$|\psi_2\rangle$$ and the control is the basis state $$| i \rangle$$of $$|\psi_1\rangle$$ with the higher $$|\alpha_i|^2$$.

Is there some primitive I can use?

• It’s not clear to me what you mean by “not measuring”. Is this a programming question? It’s also unlikely to be easy to do this efficiently. Consider the amplitudes after a single Grover iteration- the marked state has the largest amplitude, but you’d still need $O(\sqrt N)$ more iterations to have a chance to easily find it. Commented Oct 30, 2021 at 12:29
• Suppose I have two quantum registers described respectively by the quantum states $|\psi_1> = \sum_i \alpha_i |i>$ and $| \psi_2 > = | 0 >$. I would like to implement a CNOT gate where the target is $| \psi_2 >$ and the control is the state $| i >$ of $| \psi_1 >$ with the higher $| \alpha_ i|^2$. I hope this clarify my intent. Commented Nov 8, 2021 at 14:10
• that's a little clearer - can you edit your question to include this? Commented Nov 8, 2021 at 15:22

## 1 Answer

Let's consider the consequences of your proposed evolution. You would have, for example, $$(\frac35|0\rangle+\frac45|1\rangle)|0\rangle\rightarrow \frac35|00\rangle+\frac45|11\rangle$$ and also $$(-\frac45|0\rangle+\frac35|1\rangle)|0\rangle\rightarrow -\frac45|01\rangle+\frac35|10\rangle$$ Since you are discounting measurements, everything is linear. This means that we can work out what the evolution is for any other state. To help, note that this tells us $$|00\rangle\rightarrow \frac{1}{25}(9|00\rangle+16|01\rangle-12|10\rangle+12|11\rangle).$$ We can already see this is wrong. You needed $$|00\rangle\rightarrow |01\rangle$$.

So, it must be that the operation you are asking for does not exist.

• Why are you considering also the second evolution? Commented Nov 9, 2021 at 8:54
• It's not a second evolution in the sense of one after another. It's just that, by the definition of what the operation is, we can consider how it operates on multiple different inputs. I'm then using that to show a contradiction such that it cannot exist. Commented Nov 9, 2021 at 9:01
• Thank you again. Just the last clarification: how do you obtain the last evolution (the one that starts from 00) from the ones above? It is not clear to me. Commented Nov 10, 2021 at 12:26
• Well, what I should have done (although it doesn't look like I did) is 3/5 of the first equation -4/5 of the second equation. Commented Nov 10, 2021 at 12:51