When a schizophrenic man is released from a futuristic mental institution, specializing in curing your inner ?demons?, learns that he is being hunted by an assassin, he must rely on his fragmented memories to survive.

You've just identified the core of your story.

Forget backstory and setting in this instance I suggest revers engineering your concept from the goal.

Goal - Stop the man running the institute.

Now ask; why does the MC need to do this?
The answer should be as a result of a specific event that happened to the MC.

A few suggestions would be;

The man conducted an un ethical experiment on the MC he now needs revenge and wants to prevent this from happening to others.

The man killed the MC's family or friends he now needs revenge and wants to prevent this from happening to others.

The man illegally lobotomised the MC's family or friends he now needs revenge and wants to prevent this from happening to others.

There are many more options I'm sure so pick the best one and use that as the inciting incident.
Use that to start your logline then describe your character briefly and write in the goal.

What he must do is find "the man" running the institute with the help of this assassin/agent. Now how exactly do I word something like that especially since this man is one of the biggest plot points and mystery of the film.

What Nir Shelter says. To merely "survive" is not good enough. Every protagonist in every movie wants to survive. More than merely survive, the protagonist MUST need to do something, achieve some specific goal that will ensure his survival -- and defeat the antagonist.

What must the man do in this story?

Hey Sid,

When A(n) X (="troubled by his past"=one word) man is released from a futuristic institution, specializing in curing your inner demons?, [ inciting incident = "learns that he is being hunted by an agent tasked to kill him"} , [something happens=he must take action]

In other words what you really have now is:

When a schizophrenic man just released from a futuristic mental institution learns that he is being hunted by an assassin, he must....

If later in the story people fucked with his mind and he 'isn't a schizo - that's for the story to unfold...

Now that your logline is not on the front page, I would simply repost it in a few days.

Thanks everyone for the great feedback. There's lots to the story that I'm trying to condense into one sentence. Still working on it. But what do you guys think of the updated logline?

Everything said above.

Also the goal is hinted at but not specified is his goal to catch the killer? If so this needs to be specified in the logline.

Further to that the inciting incident needs to correlate to the goal in a logical line of connection. By that I mean because of the inciting incident the MC must achieve the goal therefore as suggested above the death of his family could be the inciting incident in this case.

Hope this helps.