Linear Quantum Error Correction

TL;DR

This paper develops a generalized linear map-based theory of quantum error correction that includes non-completely positive maps, impacting fault-tolerance thresholds and enabling new physical implementations.

Contribution

It introduces a comprehensive theory of quantum error correction applicable to non-CP maps, expanding the framework beyond traditional CP assumptions.

Findings

Entanglement-assisted QEC for invertible noise maps demonstrated

Non-CP recovery maps can be physically implemented

Implications for fault-tolerance thresholds in quantum computing

Abstract

We develop a generalized theory of quantum error correction (QEC) that applies to any linear map, in particular maps that are not completely positive (CP). This theory describes entanglement-assisted QEC for invertible noise maps, which we use to provides an example of the physical implementation of non-CP recovery maps. We argue that a consistent map-based theory of fault-tolerant QEC (whether Markovian or not) requires linear, non-CP maps, and that this impacts the value of the fault-tolerance threshold.