No you can't. Once Springboard is reloaded, all running apps are killed.
UPDATE: Thanks for @Mike's hint, iOS 6 introduced a new feature called State Preservation and Restoration. Read the links for more information. The App has to write handlers for preservation and restoration process in order to use this feature, and this feature is not guaranteed to have the app fully restored, as noted in the page (e.g. data model is not preserved).