A Fault-Tolerant Honeycomb Memory (2108.10457v2)

Abstract: Recently, Hastings & Haah introduced a quantum memory defined on the honeycomb lattice. Remarkably, this honeycomb code assembles weight-six parity checks using only two-local measurements. The sparse connectivity and two-local measurements are desirable features for certain hardware, while the weight-six parity checks enable robust performance in the circuit model. In this work, we quantify the robustness of logical qubits preserved by the honeycomb code using a correlated minimum-weight perfect-matching decoder. Using Monte Carlo sampling, we estimate the honeycomb code's threshold in different error models, and project how efficiently it can reach the "teraquop regime" where trillions of quantum logical operations can be executed reliably. We perform the same estimates for the rotated surface code, and find a threshold of $0.2\%-0.3\%$ for the honeycomb code compared to a threshold of $0.5\%-0.7\%$ for the surface code in a controlled-not circuit model. In a circuit model with native two-body measurements, the honeycomb code achieves a threshold of $1.5\% < p <2.0\%$, where $p$ is the collective error rate of the two-body measurement gate - including both measurement and correlated data depolarization error processes. With such gates at a physical error rate of $10^{-3}$, we project that the honeycomb code can reach the teraquop regime with only $600$ physical qubits.

Overview of "A Fault-Tolerant Honeycomb Memory"

The paper under discussion presents a significant contribution to the domain of quantum error correction through the examination of the honeycomb code, a novel quantum memory architecture. This work quantifies the efficiency and robustness of logical qubits maintained by the honeycomb code, which exploits two-local measurements and demonstrates sparse connectivity properties advantageous for certain quantum hardware designs. The paper employs Monte Carlo sampling techniques to obtain threshold estimates for the honeycomb code under various error models, comparing its performance against the well-studied rotated surface code.

Key Results and Comparative Analysis

The investigation reveals that the honeycomb code achieves lower thresholds in the controlled-not circuit model compared to the surface code, ranging from 0.2% to 0.3%, versus the surface code's 0.5%-0.7%. The significant edge, however, emerges for the honeycomb code in circuits leveraging native two-body measurements, where it achieves thresholds of 1.5% to 2.0%. In this context, the honeycomb code is projected to efficiently reach the "teraquop regime," enabling a trillion quantum operations with merely 600 physical qubits, given a physical error rate of 0.1%.

Theoretical Implications

Theoretical implications of this research are profound, as the honeycomb code exemplifies a robust subsystem code that balances locality and fault tolerance effectively. It moves beyond the conventional geometrically local stabilizer codes by integrating dynamic logical qubits that adjust with time through non-static subsystem definitions, an innovation by Hastings and Haah.

Practical Implications and Future Directions

Practically, the potential reduction in overhead for fault-tolerant quantum computing via the honeycomb code is substantial. It suggests pathways for more hardware-efficient quantum processors, particularly in platforms where two-body interactions are native operations, such as certain superconducting architectures and possibly in forthcoming Majorana-based technologies.

Future work could delve into augmenting honeycomb code architectures with boundary conditions, analyzing how these contribute to further reducing qubit overhead and error rates. There is also room to explore the rotational/shearing transformation to the lattice, which may provide a more compact realization with fewer qubits required to achieve similar logical errors.

Conclusion

Overall, the honeycomb code stands as a promising new strategy in quantum error correction, particularly in architectures dominated by direct measurement operations. The paper enriches our understanding of fault tolerance in quantum computing, providing a template for developing scalable and efficient quantum computers. The research calls for continued exploration into subsystem codes that dynamically adapt to operational demands, potentially setting new standards in topological quantum error correction.