Whether a class of quantum circuits can be efficiently simulated with a classical computer, or is provably hard to simulate, depends quite critically on the precise notion of “classical simulation”. We focus on two important notions of simulator, that we refer to as poly-boxes and EPSION-simulators and, discuss how other notions of simulation relate to these. A poly-box is a classical algorithm that outputs additive 1/poly precision estimates of Born probabilities and marginals. We present a general framework used to construct poly-boxes. This framework generalizes a number of recent works on simulation. As an application, we use the general framework to construct a classical additive 1/poly precision Born rule probability estimation algorithm for Clifford plus T circuits. Our algorithm scales exponentially in the number of T gates but polynomially in all other parameters and is intended to be state of the art for this estimation task. We expect this result to be particularly useful in the characterization and verification of near term quantum devices. We argue that the notion of classical simulation we call EPSION-simulation, captures the essence of possessing “equivalent computational power” to the quantum system it simulates: It is statistically impossible to distinguish an agent with access to an EPSION-simulator from one possessing the simulated quantum system. We relate EPSION-simulation to various alternative notions of simulation predominantly focusing on its relation to poly-boxes. Accepting some plausible computational theoretic assumptions, we show that EPSION-simulation is strictly stronger than a poly-box by showing that IQP circuits and unconditioned magic-state injected Clifford circuits are both hard to EPSION-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)
