It looks like the problems gamers are encountering with pre-loading Call of Duty: Advanced Warfare aren't limited to the Xbox One version. It appears the install problems also exist in PS4 as well.
In a tweet today, PlayStation Support acknowledged "issues with [Call of Duty: Advanced Warfare]," while directing PS4 owners to the support page where they suggest the following:
If you pre-loaded [Call of Duty: Advanced Warfare] and you're experiencing issues with the game, you must delete all pre-load files from your system and re-download all Call of Duty: Advanced Warfare content.
Please ensure you download all available updates for Call of Duty: Advanced Warfare and your system is running the latest system software.
Re-download steps
Delete Pre-Load Game Content from the Dynamic Menu.
Go to your Library and re-download the content.
I'm assuming they'll eventually come up with a more stable solution, but for now this is all Sony recommends.
Call of Duty: Advanced Warfare doesn't technically release until Tuesday, November 4; however, those who pre-ordered the Day Zero Edition were technically able to pre-load the game and begin playing a full 24 hours ahead of the official release. During this 24 hour early access, there is also double XP, which Activision/Sledgehammer Games hopefully extends given the problems with pre-loading.