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Demonstrating quantum supremacy, a complexity-guaranteed quantum advantage against over the best classical algorithms by using less universal quantum devices, is an important near-term milestone for quantum information processing. Here we develop a threshold theorem for quantum supremacy with noisy quantum circuits in the pre-threshold region, where quantum error correction does not work directly. We show that, even in such a region, we can virtually simulate quantum error correction by<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1610.03632v1">arXiv:1610.03632v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hfhatfdxsvcyxa4vhtsgefub2i">fatcat:hfhatfdxsvcyxa4vhtsgefub2i</a> </span>
more »... ction. This allows us to show that the output sampled from the noisy quantum circuits (without postselection) cannot be simulated efficiently by classical computers based on a stable complexity theoretical conjecture, i.e., non-collapse of the polynomial hierarchy. By applying this to fault-tolerant quantum computation with the surface codes, we obtain the threshold value 2.84% for quantum supremacy, which is much higher than the standard threshold 0.75% for universal fault-tolerant quantum computation with the same circuit-level noise model. Moreover, contrast to the standard noise threshold, the origin of quantum supremacy in noisy quantum circuits is quite clear; the threshold is determined purely by the threshold of magic state distillation, which is essential to gain a quantum advantage.
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