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Investigating the classical simulability of quantum circuits provides apromising avenue towards understanding the computational power of quantumsystems. Whether a class of quantum circuits can be efficiently simulated witha probabilistic classical computer, or is provably hard to simulate, dependsquite critically on the precise notion of "classical simulation" and inparticular on the required accuracy. We argue that a notion of classicalsimulation, which we call epsilon-simulation, captures the essence ofpossessing "equivalent computational power" as the quantum system it simulates:It is statistically impossible to distinguish an agent with access to anepsilon-simulator from one possessing the simulated quantum system. We relateepsilon-simulation to various alternative notions of simulation predominantlyfocusing on a simulator we call a poly-box. A poly-box outputs 1/poly precisionadditive estimates of Born probabilities and marginals. This notion ofsimulation has gained prominence through a number of recent simulabilityresults. Accepting some plausible computational theoretic assumptions, we showthat epsilon-simulation is strictly stronger than a poly-box by showing thatIQP circuits and unconditioned magic-state injected Clifford circuits are bothhard to epsilon-simulate and yet admit a poly-box. In contrast, we also showthat these two notions are equivalent under an additional assumption on thesparsity of the output distribution (poly-sparsity).

From estimation of quantum probabilities to simulation of quantum circuits