  Extended Clifford circuits straddle the boundary between classical and
quantum computational power. Whether such circuits are efficiently classically
simulable seems to depend delicately on the ingredients of the circuits. While
some combinations of ingredients lead to efficiently classically simulable
circuits, other combinations, which might just be slightly different, lead to
circuits which are likely not. We extend the results of Jozsa and Van den Nest
[Quant. Info. Comput. 14, 633 (2014)] by studying two further extensions of
Clifford circuits. First, we consider how the classical simulation complexity
changes when we allow for more general measurements. Second, we investigate
different notions of what it means to "classically simulate" a quantum circuit.
These further extensions give us 24 new combinations of ingredients compared to
Jozsa and Van den Nest, and we give a complete classification of their
classical simulation complexities. Our results provide more examples where
seemingly modest changes to the ingredients of Clifford circuits lead to
"large" changes in the classical simulation complexities of the circuits, and
also include new examples of extended Clifford circuits that exhibit "quantum
supremacy", in the sense that it is not possible to efficiently classically
sample from the output distributions of such circuits, unless the polynomial
hierarchy collapses.
