In my experience, unless the problem is very intermittent (e.g. once a year), Err01 is almost always caused by a bad ribbon cable in the lens itself—usually the aperture ribbon.

The aperture ribbon, because of where the aperture is located, can get knocked out of position if the lens gets bumped in the wrong way, and in some situations, this can lead to the edge of the ribbon getting snagged on other moving parts inside the lens. When this happens, the thin ribbon cable tears, and the lens cannot change its aperture.

When the camera tells the lens to change its aperture, the lens tells the aperture motor to move, and then waits for a signal from the aperture motor to tell it that it is in the right position. If either of those two things doesn't happen, the lens doesn't tell the camera that it is in the right position, and the camera gives an Err01 indicating that it cannot communicate with the lens (when, in fact, the lens got the message).

This is, in some ways, a defect in Canon's communication protocol. The lens should respond with a failure code after a period of time so that the camera won't lock up. It is also, in some ways, a defect in Canon's lens design; the ribbon cable in question should be fastened more securely and in more places to ensure that it cannot ever move in a way that would allow it to get snagged.

Either way, the lens is toast, and the only thing you can do is get it repaired or replace it.