In the city of the future, a young man takes on a series of mind-bending, gladiatorial games to win back the woman he loves. If he dies, she suffers; but to survive, he’ll have to break the city’s only rule.

Agreed with jamesmichael, with the one addition that you should try to find a better way to describe your protagonist than "young man". Is he a rebellious hoodlum? A naive undergrad? Don't waste that word count.

I really like this idea. I think it would benefit though if you change the way it's written, only slighty.

A logline is about making the story as clear as possible. To do this I suggest writing it as a series of causes and events. So with this logline I would re-word it like so:

In a city of the future, when the woman he loves is taken by (whoever) a young man must take on a series of mind bending, gladiatorial games to win her back or risk losing her forever."

Obviously rather than the 'risk loseing her forever' bit, i would make the stakes a little clearer.

And i dont think that you need the 'breaking the city's one rule' line. This just makes the logline more confusing and makes it read more like a tagline.