Roads towards fault-tolerant universal quantum computation (1612.07330v2)

Abstract: Current experiments are taking the first steps toward noise-resilient logical qubits. Crucially, a quantum computer must not merely store information, but also process it. A fault-tolerant computational procedure ensures that errors do not multiply and spread. This review compares the leading proposals for promoting a quantum memory to a quantum processor. We compare magic state distillation, color code techniques and other alternative ideas, paying attention to relative resource demands. We discuss the several no-go results which hold for low-dimensional topological codes and outline the potential rewards of using high-dimensional quantum (LDPC) codes in modular architectures.

• The paper presents comprehensive methods for achieving fault-tolerant quantum computation using advanced error correction techniques like magic state distillation and topological codes.
• The paper emphasizes designing architectures with high noise thresholds and efficient error management to maintain logical qubit integrity during processing.
• The paper outlines promising future research avenues including higher-dimensional codes and quantum LDPC codes to enhance scalability and reliability.

Fault-Tolerant Universal Quantum Computation: A Review

The paper, "Roads towards fault-tolerant universal quantum computation," is a comprehensive review that addresses the transition from quantum memory to processing in terms of fault-tolerant quantum computations. The authors, Campbell, Terhal, and Vuillot, dissect various methodologies for achieving error-resilient quantum computation, focusing on how logical qubits can be effectively managed in the context of current experimental technologies and theoretical advancements.

Quantum Error Correction Architectures

The paper outlines the necessity of quantum computers to not only store information but also to process it while mitigating errors. Quantum error correction (QEC) is pivotal in this regard, ensuring that errors do not proliferate throughout quantum computations. The authors highlight a few essential criteria for effective quantum error correction architectures:

1. High Noise Threshold: The architecture must offer an error threshold that provides logical qubits with lower error probabilities than the physical qubits from which they are built.
2. Universal Logical Gate Set: It should support the implementation of a universal set of logical gates.
3. Spatial and Temporal Efficiency: Achieving high noise thresholds and universal gate sets must come with minimal spatial and temporal overhead.
4. Fast Error Processing: The processing and correction of errors should keep pace with quantum computational operations.

Proposed Methods

The paper discusses several leading proposals, including magic state distillation, color code techniques, and others, analyzing resource demands and other trade-offs:

• Magic State Distillation (MSD): This process involves using noisy magic states and distilling them into higher-quality states. The work analyzes the efficiencies of different distillation protocols and emphasizes their importance in achieving fault tolerance.
• Topological Codes: The authors review low-dimensional topological codes, including the surface code and color codes, noting the promising noise thresholds and resource overheads. Surface codes, with a 2D architecture and a noise threshold between 0.6% and 1%, are particularly emphasized for their practicality.
• Color Codes: Highlight the ability to support Clifford gates transversally and discuss the potential for using high-dimensional codes for correcting errors and implementing transversal gates.

Dimensionality and Transversality

A significant discussion point is the relationship between code dimensionality and the ability to implement transversal gates, as shown by Bravyi and Koenig's theorem. Higher-dimensional codes, such as 4D and 3D codes, offer potential advantages, including the possibility of single-shot error correction, demonstrated in 3D gauge color codes.

Implications and Future Research

While the combination of surface codes and magic state distillation currently sets a benchmark for fault-tolerant quantum computation, emerging architectures continue to offer alternative benefits and trade-offs. These alternatives, such as 3D color codes or even more exotic higher-dimensional LDPC codes, while not yet surpassing surface codes and MSD in practicality, suggest avenues for future research. Particularly, integrating more general quantum LDPC codes, supported by modular quantum hardware architectures, could provide avenues to leverage the full potential of quantum computation.

The paper anticipates that continued development of these approaches will enhance our understanding of robust, large-scale quantum computation. This extends an invitation to the quantum information community to further investigate and innovate along these various promising paths. The integration of advanced quantum error correction codes with scalable quantum hardware remains a critical target for achieving scalable, universal quantum computation.