# How to transform a state with amplitude squared or to any power?

Suppose I have an unknown state $$|\psi\rangle = \sum_i \alpha_i|{\lambda_i}\rangle$$, is it possible that I can transform it into $$|\psi\rangle = \frac{1}{\sqrt{\sum_i|\alpha_i|^{2r}}} \sum_i \alpha_i^r|{\lambda_i}\rangle$$?

I have an idea for one qubit with a measurement, which would be better without measurements.

Suppose the input state is $$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$$ and can be prepared with two copies. An ancilla qubit is provided with state $$|0\rangle$$, such that

$$(\alpha|0\rangle+\beta|1\rangle)(\alpha|0\rangle+\beta|1\rangle)|0\rangle= \alpha^2|000\rangle + \alpha\beta|010\rangle+\beta\alpha|100\rangle+\beta^2|110\rangle.$$

With two CNOT gates in a row, the ancilla qubit is the target qubit, such that

$$\alpha^2|000\rangle+\alpha\beta|011\rangle+\beta\alpha|101\rangle+\beta^2|110\rangle.$$

This is followed by a measurement on ancilla qubit if we happen to measure 0, which the state on the first two qubits will be $$\frac{\alpha^2}{\sqrt{|\alpha|^4+|\beta|^4}}|000\rangle+\frac{\beta^2}{\sqrt{|\alpha|^4+|\beta|^4}}|110\rangle.$$

With an CNOT gate on the second qubit, using the first qubit as control, such that

$$\frac{\alpha^2}{\sqrt{|\alpha|^4+|\beta|^4}}|00\rangle+\frac{\beta^2}{\sqrt{|\alpha|^4+|\beta|^4}}|10\rangle= (\frac{\alpha^2}{\sqrt{|\alpha|^4+|\beta|^4}}|0\rangle+\frac{\beta^2}{\sqrt{|\alpha|^4+|\beta|^4}}|1\rangle)|0\rangle$$

The state in the first qubit will be

$$\frac{\alpha^2}{\sqrt{|\alpha|^4+|\beta|^4}} |0\rangle+\frac{\beta^2}{\sqrt{|\alpha|^4+|\beta|^4}} |1\rangle$$

However, the measurement on ancilla qubit is a nuisance. Can I obtain the powered amplitude state without measurement on arbitrary number of qubits?

• This is not possible. This powering transformation you're trying to implement is non-linear, so it can't be done via unitary transformations, which are always linear. The only way to do it is probabilistically via measurements, as you did. It's anyway an interesting question what is the optimal way of doing it. Jul 18, 2020 at 9:21
• @MateusAraújo Yeah, you're right. As the power increases, leave alone the feasibility for more qubits, the chance to obtain such a state diminishs exponentially. Jul 18, 2020 at 10:06

Also note at, as $$r\to\infty$$, the transformation becomes a projection onto the subspace spanned by the states with highest modulus. You do are doing something like a weak measurement when $$r$$ is finite.
Nearly final observation: you mention you want $$|\psi\rangle$$ to be "unknown". You should be cautious taking your solution (as you generalise requiring more copies of $$|\psi\rangle$$) farther without thinking about no-cloning or more subtle resource counting.