I'm trying to implement (simulate) the Novel Enhanced Quantum Representation (NEQR), which is one of the quantum image representation models, but i'm stuck in the measurement part. In other words i don't know how to retrieve the original image from the quantum state.

The image is represented by the following formula (this is the final form after preparation): $$ \left| \psi \right\rangle = \frac{1}{2^n} \displaystyle\sum_{Y=0}^{2^n -1} \displaystyle\sum_{X=0}^{2^n-1} \left| f(Y,X) \right\rangle \left| YX \right\rangle $$

Now, assume that i have a 2 x 2 image and the color of each pixel is 2 bit only, the position -> color cobinations are as follow:

YX -> color
00 -> 11
01 -> 00
10 -> 01
11 -> 00

The image or the final state which is composed of two entangled qubit sequences (color and position) should be a 16 x 1 column vector indexed by binary representation from 0 to 15 (e.g. 0000, 0001, ... 1111) and the value of 0001 will be 1 because color 00 is located in position 01.

The measurement is as follow:

During image retrieval from the quantum image, every pixel should be recovered individually. The following operations have been devised to retrieve pixel from a $$ 2^n × 2^n $$ quantum image.

How to perform measurement on the quantum state to retrieve the original image. In other words how to construct the matrix that when i multiply it with the state i retrieve the original image, but how to do that according to Quantum Operations ?

Please help to to solve this issue or guide me to the correct path. I didn't add all equations because i don't know if that is legal so please look for them in the paper.



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