What is the simplest autotrophic cell?
If by "simplest" you mean smallest genome, then according to Brock's Biology of Microrganisms (13th ed, p 319) the smallest autotroph is Aquifex aeolicus, a chemolithotroph bacterium. It's genome is 1.5M bases, about 1/3 of that of E.Coli, and was sequenced in 1998 (https://www.ncbi.nlm.nih.gov/pubmed/9537320).

The genome is densely packed though and still carries some 1,500 open reading frames. The authors of the genome sequencing paper states that "metabolic flexibility seems to be reduced as a result of the limited genome size", which I guess is some evidence of "simplicity".