Bosons vs. Fermions – A computational complexity perspective

Recent years have seen a flurry of activity in the fields quantum computing and complexity theory, which aim to understand computational capabilities systems by applying toolbox theory. This paper explores conceptually rich technologically useful connection between dynamics free particles I review results on power two simple systems, built out noninteracting bosons (linear optics) or fermions. These rudimentary computers display radically different capabilities—while fermions are easy simulate classical computer, therefore devoid nontrivial power, free-boson computer can perform tasks expected be classically intractable. To build argument for these results, introduce concepts from describe some classes, starting with P NP building up less common #P polynomial hierarchy, relations them. identify how probabilities free-bosonic free-fermionic fit within this classification, then underpins their difference power. is aimed at graduate advanced undergraduate students Physics background, hopefully serving as soft introduction exciting highly evolving field.