Does the iPhone X use facial recognition hardware? Does the iPhone X use Facial Recognition hardware not found in previous models? Or is the algorithm executed on the CPU?
Both. The iPhone X's True Depth Camera uses hardware unique to the X, specifically an infrared camera, an IR flood illuminator, and a dot projector to measure the depth of your face. The regular FaceTime camera is used for color information and iOS software handles facial recognition, but not concerning the "fingerprint" of your face for Face ID - facial detection works in realtime on all current iPhones and facial recognition exists in the photos app via software.

The True Depth camera does all the heavy lifting for Face ID. Additionally, Face ID works in the dark because of the IR camera, flood illuminator, and dot projector. Without this hardware, the regular FaceTime camera could be fooled by a sheet of paper with a person's face on it.