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

Question

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?

Answer 1

As noted by Mateus in the comments, the transformation you are looking for is non-linear. This cannot be done with any matrix transformation. Thus, you will need more qubits, and your solution shows two (+1 scratch qubit) is sufficient. I guess you might wonder if a two-qubit unitary can do it, though?

The problem is that the transformation you want to implement depends on the input state. You can't do this (unitarily) even with extra qubits. I believe the most general result forbidding such requirements is the No-Programming Theorem.

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.

Last thing. A coherent version of something like what you might be looking for is Amplitude Amplification.