>To exist the compressed file only needs a formula to validate the result. Like it being a type 42 image with this checksum. It might take a quantum computer to confirm existence

I interpret this to mean that the checksum is the compressed file?

Surely you'd need to compute checksums for all possible images to confirm that the checksums are unique.

You're calculating a perfect hash for an input that could be anything. That isn't going to be short or easy to confirm.

And then you've got the question of if it's worth it. Yes the hash could represent an image of random noise at the same size as a blank black image. But it's going to be larger than today's technology for representing that black image.

So for a lot of images it isn't worth it. I would guess(?) the average photo is a lot closer to ordered than unordered, so it isn't going to work there either.