Noise Threshold of Quantum Supremacy

TL;DR

This paper establishes a noise threshold for quantum supremacy in noisy quantum circuits, demonstrating that even without full error correction, quantum advantage can be maintained below a certain error rate, with a threshold of 2.84%.

Contribution

The authors develop a threshold theorem for quantum supremacy in the pre-threshold noise region, linking the supremacy threshold to magic state distillation.

Findings

Threshold for quantum supremacy is 2.84%, higher than standard fault-tolerance thresholds.

Quantum supremacy can be virtually simulated via postselection even without direct error correction.

The supremacy threshold is determined by the magic state distillation threshold.

Abstract

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 postselection. 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.