I want today to be my last day in my job. Can I give my resignation letter tomorrow? [Ed: without notice]
You CAN. I wouldn't recommend it, though.
Will there be a problem? And what if I can't provide the 30-day notice?
Depending on your location and local laws, you might get sued for breach of contract. Even if the company doesn't care enough to follow through on that, it will obviously impact your future efforts to find a new job. Nobody wants to hire a guy who doesn't respect the terms of his contracts and might suddenly quit without giving notice.
If you really want/need to leave but cannot give the full 30 days notice as required by the contract, then you should at least give as much notice as possible. Speak to your manager, explain that you want to quit but can only give X days notice. There's a good chance they'll accept it, even if they aren't happy about it, because you're being honest and upfront about the situation.
Also, if you've only been there a week, you may still be within the probationary period (again, depending on your location) and therefore not actually be required to give the full 30 days notice. Again, check your contract for probationary periods and speak with your manager.
Whatever you do, don't try to pull a fast one and just walk out without explaining yourself or trying to talk with people. That's only going to piss them off and ruin your own reputation in the process.