Quantum Computing Asked by M. Chen on August 20, 2021

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

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

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

$

(alpha|0rangle+beta|1rangle)(alpha|0rangle+beta|1rangle)|0rangle=

alpha^2|000rangle + alphabeta|010rangle+betaalpha|100rangle+beta^2|110rangle.

$

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

$

alpha^2|000rangle+alphabeta|011rangle+betaalpha|101rangle+beta^2|110rangle.

$

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}}|000rangle+frac{beta^2}{sqrt{|alpha|^4+|beta|^4}}|110rangle.

$

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

$

frac{alpha^2}{sqrt{|alpha|^4+|beta|^4}}|00rangle+frac{beta^2}{sqrt{|alpha|^4+|beta|^4}}|10rangle=

(frac{alpha^2}{sqrt{|alpha|^4+|beta|^4}}|0rangle+frac{beta^2}{sqrt{|alpha|^4+|beta|^4}}|1rangle)|0rangle

$

The state in the first qubit will be

$

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

$

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

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 $rtoinfty$, 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 $|psirangle$ to be "unknown". You should be cautious taking your solution (as you generalise requiring more copies of $|psirangle$) 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.

Correct answer by Chris Ferrie on August 20, 2021

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