Fault-tolerant complexes

TL;DR

This paper introduces a geometric framework called fault-tolerant complexes for designing and analyzing surface-code fault-tolerant quantum protocols, enabling automated discovery of high-threshold schemes and extending homological fault tolerance descriptions.

Contribution

It defines fusion complexes and fault-tolerant complexes, providing a systematic, geometric approach to identify and analyze new fault-tolerant quantum computing schemes.

Findings

Identified 627 new fault-tolerance schemes via automated search.

Found fusion networks with higher erasure and Pauli thresholds than existing schemes.

Extended homological fault tolerance to a broader class of error models.

Abstract

Fault-tolerant complexes describe surface-code fault-tolerant protocols from a single geometric object. We first introduce fusion complexes that define a general family of fusion-based quantum computing (FBQC) fault-tolerant quantum protocols based on surface codes. We show that any 3-dimensional cell complex where each edge has four incident faces gives a valid fusion complex. This construction enables an automated search for fault tolerance schemes, allowing us to identify 627 examples within a moderate search time. We implement this using the open-source software tool Gavrog and present threshold results for a variety of schemes, finding fusion networks with higher erasure and Pauli thresholds than those existing in the literature. We then define more general structures we call fault-tolerant complexes that provide a homological description of fault tolerance from a large family of low-level error models, which include circuit-based computation, floquet-based computation, and FBQC with multi-qubit measurements. This extends the applicability of homological descriptions of fault tolerance, and enables the generation of many new schemes which have not been previously identified. We also define families of fault-tolerant complexes for color codes and 3d single-shot subsystem codes, which enables similar constructive methods, and we present several new examples of each.