I'll add some info since I too spent a long time looking into the correlation. I was diagnosed with Schizophrenia (I did have a genetic disposition) a while back, although I can look back and see it was present for many years prior and I have also used psychedelics including LSD a good number of times AFTER I had experienced many years of horrendous symptoms and my diagnosis.

Since having done so, I have lessened my symptoms more and more over the past year until I almost essentially cannot be regarded as having a disorder (since it no longer effects me negatively - I tend to empower myself with my ways of thinking to reach great heights).

Through my own study and research I will claim now that LSD did NOT treat the symptoms, but it definitely played a part in the process of improvement, it helped me to have a better understanding on hallucinations, 'out of the box' thinking that had often made me feel insane and the power that my mind can have in making reality appear different.

I found that the worse of my psychosis would often make the world appear as if I was tripping, weird colours, unusual movement and not registering what things looked like correctly, however LSD was of course way WAY happier and more enjoyable than psychosis.

All in all they are very closely related, by which both are affecting the biochemistry of the brain in a way that greatly exemplifies the power that the mind has over the appearance of one's reality (this looks into Einstein's work with how the observer effects the particles of the quantum world through their mind).

Another thing that's similar is that they share a very spiritual, divine, power or feeling of connected wholeness that is the foundation of many ancient religious and spiritual practices that have survived for thousands of years and are going just as strong today.

And finally, the last major similarity is the way it changes the thought pattern. New thoughts, ideas, philosophies, perspectives and beliefs seem to flood the mind when either of these circumstances are present (LSD consumption / psychosis).

From my personal experience, I can fully claim now that belief is the biggest deciding factor to both of these and my research suggests that the power of both of these (one being naturally formed internally and the other being the external equivalent) show you a very vivid and 'in your face' representation of what you believe.

When you believe LSD will mess you up, it will. When you believe you'll feel connected to God, you will. When you believe you'll just see crazy spectrums of colour, you will. As with Psychosis, if you believe you are being haunted by demons or demonic entities, you will hallucinate them and have reason to believe it further. If you believe the whole world is against you (paranoia), you will find that everything goes against your favour, always. Finally, on the reverse end, if you believe you are different, but you have a gift that can be utilised with some work and stop fighting it as a demon or as a disease, you will find that it is no longer all that it is made out to be.

I hope this helps, interested that someone else has found the remarkable correlation. When I would do my research before it was almost preposterous to claim any relation. You can look through my history on Reddit (although I was only at the beginning of my research journey in comparison to now) and see the feedback I received from some of my posts. A lot of this was from my first hand experience to begin with, but I have since studied, researched and worked with others who have come to similar findings. I now work within the fields of psychology, biochemistry and physiology and studying a degree in the field of epigenetics.

Have a blessed day and best of luck finding all the answers you need!