Fault tolerance of stabilizer channels

TL;DR

This paper develops a comprehensive formalism to analyze the fault tolerance of stabilizer channels under various noise models, enabling improved fault-tolerant circuit design for quantum error correction.

Contribution

It introduces a rigorous framework for fault tolerance analysis of stabilizer channels, including new definitions, algorithms, and the concept of hook faults, applicable to diverse quantum codes.

Findings

Framework applied to surface codes reveals new fault-tolerant circuit features

Defines generalized hook faults for flexible fault-tolerance analysis

Establishes conditions for fault distance preservation under channel composition

Abstract

Stabilizer channels are stabilizer circuits that implement logical operations while mapping from an input stabilizer code to an output stabilizer code. They are widely used to implement fault tolerant error correction and logical operations in stabilizer codes such as surface codes and LDPC codes, and more broadly in subsystem, Floquet and space-time codes. We introduce a rigorous and general formalism to analyze the fault tolerance properties of any stabilizer channel under a broad class of noise models. This includes rigorous but easy-to-work-with definitions and algorithms for the fault distance and hook faults for stabilizer channels. The generalized notion of hook faults which we introduce, defined with respect to an arbitrary subset of a circuit's faults rather than a fixed phenomenological noise model, can be leveraged for fault-tolerant circuit design. Additionally, we establish necessary conditions such that channel composition preserves the fault distance. We apply our framework to design and analyze fault tolerant stabilizer channels for surface codes, revealing novel aspects of fault tolerant circuits.