The Role of Correlated Noise in Quantum Computing

TL;DR

This paper surveys fault-tolerant quantum computing, focusing on noise models, thresholds, and limitations, highlighting recent results on correlated and non-Markovian noise and their implications for quantum error correction.

Contribution

It provides a comprehensive overview of threshold results for various noise models, including correlated and non-Markovian noise, and discusses limitations in quantum error correction.

Findings

Thresholds exist for independent and certain correlated noise models.

Non-Markovian noise can still allow for fault-tolerance under specific conditions.

There are fundamental limitations to error correction in quantum systems.

Abstract

This paper aims to give an overview of the current state of fault-tolerant quantum computing, by surveying a number of results in the field. We show that thresholds can be obtained for a simple noise model as first proved in [AB97, Kit97, KLZ98], by presenting a proof for statistically independent noise, following the presentation of Aliferis, Gottesman and Preskill [AGP06]. We also present a result by Terhal and Burkard [TB05] and later improved upon by Aliferis, Gottesman and Preskill [AGP06] that shows a threshold can still be obtained for local non-Markovian noise, where we allow the noise to be weakly correlated in space and time. We then turn to negative results, presenting work by Ben-Aroya and Ta-Shma [BT11] who showed conditional errors cannot be perfectly corrected. We end our survey by briefly mentioning some more speculative objections, as put forth by Kalai [Kal08, Kal09, Kal11].