# 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. Oct 30 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. Nov 8 at 14:10
• that's a little clearer - can you edit your question to include this? Nov 8 at 15:22

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 $$|00\rangle\rightarrow |01\rangle$$.