You could certainly open a restaurant in the U.S. as a Canadian citizen. No financial or identification documents would be required by the states, nor would you be required to provide a social security number, and you may list your company in Canada as the owner of the U.S. company.
You would need to have the company registered (as an LLC or C-Corporation) with the state where the restaurant would be physically located, and obtain at the least the federal tax id (EIN) as well as that state's sales/use tax permit, where applicable.
Of course, before doing all that you should decide on how this restaurant would be managed. If you plan to run it personally you should consult an immigration attorney to see how it can be achieved from legal point of view.