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
Get help from others!
Recent Questions
Recent Answers
© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP