Further extensions of Clifford circuits and their classical simulation complexities

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.