Fault-tolerant complexes (2308.07844v1)

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.

• The paper introduces a comprehensive homological framework for designing fault-tolerant protocols in quantum computing.
• It leverages fusion complexes to automate the discovery of 627 viable fault-tolerant schemes via advanced geometric methods.
• Efficacy is demonstrated with Pauli error thresholds above 1% and erasure thresholds nearing 12%, enhancing computation reliability.

Fault-Tolerant Complexes in Quantum Computing

The paper "Fault-tolerant complexes" presents a comprehensive framework for designing and analyzing fault-tolerant protocols within quantum computing architectures, predominantly focused on fusion-based quantum computing (FBQC). By introducing the concept of fusion complexes and extending it to fault-tolerant complexes, the research takes significant strides in developing a robust homological framework applicable to various quantum error correction schemes.

Fusion Complexes and Fault-Tolerant Complexes

The paper begins with the introduction of fusion complexes, which are specifically structured to support surface-code fault-tolerant protocols. These complexes leverage the geometry of three-dimensional cell complexes, ensuring that each edge has four incident faces, permitting the construction of extensive fault-tolerant schemes. This geometric representation is pivotal as it facilitates the automated generation of fault tolerance schemes, evidenced by the identification of 627 examples using the Gavrog software tool.

The authors further generalize this notion to fault-tolerant complexes, which describe a broader set of fault tolerance protocols beyond those constructed via fusion-based models. These complexes provide a homological description for fault tolerance applicable to multiple error models, including circuit-based computation, floquet-based computation, and FBQC with multi-qubit measurements. This generalization not only broadens the applicability of their framework but also enables the emergence of novel fault-tolerant schemes heretofore unexplored.

Numerical Results and Performance

Multiple examples of fusion complexes are illustrated, and threshold results are computed portraying their efficacy. Notably, the paper reports fusion networks with enhanced erasure and Pauli thresholds compared to existing literature, with some complexes exhibiting thresholds higher than 1% for Pauli errors and approximately 12% for erasure errors. These robust numerical results showcase the potential of the framework in improving the reliability and scalability of quantum computing systems.

Theoretical and Practical Implications

The theoretical implications of this research are profound, as it extends the homological approach to a much larger class of error models, establishing a fundamental link between the size and structure of computational primitives and the configuration of check operators. Practically, the ability to automate the search for new fault-tolerant schemes is invaluable, offering a pathway to accelerate the development of more efficient and high-performing quantum algorithms.

Moreover, the introduction of fault-tolerant complexes facilitates a unified view of various computational models, potentially serving as a bridge between disparate error-correction strategies. This framework may inspire future explorations into adapting these concepts to other quantum computing models or integrating them into existing error correction paradigms to enhance their resilience.

Future Prospects

The potential future developments that arise from this work could include an in-depth exploration of the relationships between different computational models using fault-tolerant complexes, the adaptation of these complexes to more exotic error models, or even to non-photonic systems. Additionally, extending the use of automated tools like Gavrog to encompass a wider array of quantum systems and error-correction protocols can unlock new insights into quantum fault tolerance.

In conclusion, the paper provides a substantial contribution to quantum error correction frameworks, demonstrating both methodological rigor and innovative potential. By facilitating a more structured and comprehensive approach to realizing fault tolerance in quantum computing, it sets the stage for future advancements in the field.