From estimation of quantum probabilities to simulation of quantum circuits

TL;DR

This paper explores the classical simulation of quantum circuits, introducing epsilon-simulation as a measure of computational equivalence and analyzing its relationship with other simulation notions like poly-box, revealing complexity distinctions.

Contribution

It defines epsilon-simulation as a robust notion of classical simulation and compares it to poly-box, establishing complexity results for IQP and Clifford circuits under various assumptions.

Findings

Epsilon-simulation is strictly stronger than poly-box under certain assumptions.

IQP and magic-state Clifford circuits are hard to epsilon-simulate but admit a poly-box.

Under sparsity assumptions, epsilon-simulation and poly-box are equivalent.

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, 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).