  Investigating the classical simulability of quantum circuits provides a
promising avenue towards understanding the computational power of quantum
systems. Whether a class of quantum circuits can be efficiently simulated with
a probabilistic classical computer, or is provably hard to simulate, depends
quite critically on the precise notion of "classical simulation" and in
particular on the required accuracy. We argue that a notion of classical
simulation, which we call epsilon-simulation, captures the essence of
possessing "equivalent computational power" as the quantum system it simulates:
It is statistically impossible to distinguish an agent with access to an
epsilon-simulator from one possessing the simulated quantum system. We relate
epsilon-simulation to various alternative notions of simulation predominantly
focusing on a simulator we call a poly-box. A poly-box outputs 1/poly precision
additive estimates of Born probabilities and marginals. This notion of
simulation has gained prominence through a number of recent simulability
results. Accepting some plausible computational theoretic assumptions, we show
that epsilon-simulation is strictly stronger than a poly-box by showing that
IQP circuits and unconditioned magic-state injected Clifford circuits are both
hard to epsilon-simulate and yet admit a poly-box. In contrast, we also show
that these two notions are equivalent under an additional assumption on the
sparsity of the output distribution (poly-sparsity).
