A Center Controller reported an aircraft on a SID turned direct to their destination early and flew below the Minimum IFR Altitude.

Narrative

I was training Person 1 during this session. Aircraft X departed ZZZ Runway X and was on the SID climbing to 14;000 ft. Person 1 radar contacted the aircraft and asked if he was able 16;000 ft. The pilot responded affirmative; and Person 1 responded with 'Aircraft X; Climb and maintain 16;000 ft. and reaching 16;000 ft. cleared left hand turn direct ZZZ1. We thought the pilot read back the clearance correctly; and we issued weather and proceeded to move on working the sector. A couple minutes later Person 1 noticed Aircraft X made a left hand turn direct into higher terrain. Person 1 issued him a low altitude alert. Then I came on frequency and told him to suggest he turns right immediately to a 020 heading.Then he entered the higher terrain MIA and I suggested a 020 heading again immediately. He made the turn and then climbed above terrain. Then we issued him a 040 heading to keep him above terrain. When he was above the MIA; we gave him back on course. Person 1 issued the brasher warning and shipped the aircraft. We watched the Aircraft X and the initial clearance was correct. However; he read back 'climb and maintain 16;000 cleared left turn direct destination'. So unfortunately; we missed that bad readback. Bad readbacks are a part of the job and it happens; and we do our best to listen and our best to catch them. Person 1 and I both missed this one.