  The study of quantum circuits composed of commuting gates is particularly
useful to understand the delicate boundary between quantum and classical
computation. Indeed, while being a restricted class, commuting circuits exhibit
genuine quantum effects such as entanglement. In this paper we show that the
computational power of commuting circuits exhibits a surprisingly rich
structure. First we show that every 2-local commuting circuit acting on d-level
systems and followed by single-qudit measurements can be efficiently simulated
classically with high accuracy. In contrast, we prove that such strong
simulations are hard for 3-local circuits. Using sampling methods we further
show that all commuting circuits composed of exponentiated Pauli operators
e^{i\theta P} can be simulated efficiently classically when followed by
single-qubit measurements. Finally, we show that commuting circuits can
efficiently simulate certain non-commutative processes, related in particular
to constant-depth quantum circuits. This gives evidence that the power of
commuting circuits goes beyond classical computation.
