From estimation of quantum probabilities to simulation of quantum circuits

Hakop Pashayan1, Stephen D. Bartlett1, and David Gross2

1Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia
2Institute for Theoretical Physics, University of Cologne, Germany

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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 (or $\epsilon$-simulation for short), 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 $\textit{poly-box}$. 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 ($\textit{poly-sparsity}$).

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