So a few updates ago there was a problem with cloud sync which was draining the battery fast! They have since fixed it with updates so update to iOS 16.3.1 which should fix the problem you’re having. Below is the original problem discussion.
Thread 'Contacts Battery Drain'
https://forums.macrumors.com/threads/contacts-battery-drain.2371195/
Thread 'Contacts Battery Drain'
https://forums.macrumors.com/threads/contacts-battery-drain.2371195/