Abstract
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 (orϵ-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 anϵ-simulator from one possessing the simulated quantum system. We relateϵ-simulation to various alternative notions of simulation predominantly focusing on a simulator we call apoly-box. A poly-box outputs1/polyprecision 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 thatϵ-simulation is strictly stronger than a poly-box by showing that IQP circuits and unconditioned magic-state injected Clifford circuits are both hard toϵ-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).
Highlights
Introduction and summary of main resultsWhich quantum processes can be efficiently simulated using classical resources is a fundamental and longstanding problem [1, 2, 3, 4, 5, 6]
Works on boson sampling [7], instantaneous quantum polynomial (IQP) circuits [8, 9], various translationally invariant spin models [10, 11], quantum Fourier sampling [12], one clean qubit circuits [13, 14], chaotic quantum circuits [15] and conjugated Clifford circuits [16] have focused on showing the difficulty of classically simulating these quantum circuits
There has been substantial recent progress in classically simulating various elements of quantum systems including matchgate circuits with generalized inputs and measurements [17], circuits with positive quasi-probabilistic representations [19, 20, 21], stabilizer circuits supplemented with a small number of T gates [22], stabilizer circuits with small coherent local errors [23], noisy IQP circuits [24], noisy boson sampling circuits [25], low negativity magic state injection in the fault tolerant circuit model [26], quantum circuits with polynomial bounded negativity [27], Abelian-group normalizer circuits [28, 29] and certain circuits with computationally tractable states and sparse output distributions [30]
Summary
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 -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 -simulator from one possessing the simulated quantum system. 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. We show that these two notions are equivalent under an additional assumption on the sparsity of the output distribution (poly-sparsity)
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