Single-shot and measurement-based quantum error correction via fault complexes

TL;DR

This paper introduces the fault complex framework to analyze measurement-based quantum error correction, improving understanding and thresholds of foliated codes in photonic quantum computing.

Contribution

The paper presents the fault complex formalism, enabling detailed analysis of fault tolerance and single-shot error correction in measurement-based quantum codes.

Findings

Improved fault tolerance thresholds for 3D and 4D toric codes

Generalization of stability experiments

Existence of single-shot lattice surgery with higher-dimensional codes

Abstract

Photonics provides a viable path to a scalable fault-tolerant quantum computer. The natural framework for this platform is measurement-based quantum computation, where fault-tolerant graph states supersede traditional quantum error-correcting codes. However, the existing formalism for foliation - the construction of fault-tolerant graph states - does not reveal how certain properties, such as single-shot error correction, manifest in the measurement-based setting. We introduce the fault complex, a representation of dynamic quantum error correction protocols particularly well-suited to describe foliation. Our approach enables precise computation of fault tolerance properties of foliated codes and provides insights into circuit-based quantum computation. Analyzing the fault complex leads to improved thresholds for three- and four-dimensional toric codes, a generalization of stability experiments, and the existence of single-shot lattice surgery with higher-dimensional topological codes.