Corporate pilot reported descending below a crossing altitude on approach prior to the fix resulted in an ATC low altitude alert. Pilot performed a missed approach.

Narrative

While being vectored for the RNAV (GPS) Runway XX at ZZZ I had been descended to 3000 MSL. I was cleared direct to an IF called ZZZZZ. Shortly after that I was cleared for the approach and instructed to 'maintain 3000 ft. until established cleared the RNAV XX ZZZ'. At this point in time I was only a few miles from ZZZZZ with the autopilot on and all appropriate selections on the flight guidance panel in order to fly the approach including APPR mode and lastly with the LPV DA set. The weather was at minimums and I anticipated a likely possible missed approach. I reviewed the missed approach procedure on the approach plate a second time just in case I had to miss. I then scanned my instruments; and noticed that the autopilot had started to descend prior to ZZZZZ and was to the best of my recollection at 2700 ft. I realized and started to correct and at the same moment; ATC queried with a 'low altitude alert'. I continued to correct; executed the approach; had to miss and was then vectored for the RNAV XY and was able to complete this approach and land. There was never a terrain warning in the airplane nor any loss of separation.In retrospect; I think that I may have taken my eyes off the instrument panel a little too long to review the missed approach procedure. The takeaway is to be sure that all aspects of the approach are reviewed well in advance and when close to commencing the approach; trust but verify the autopilot and what it is doing. I am not sure why the autopilot did what it did; because I believe that I had the approach phase functions properly selected. I will be more vigilant on scanning and trusting the autopilot in the future.