Improved single-shot decoding of higher dimensional hypergraph product codes

Abstract: In this work we study the single-shot performance of higher dimensional hypergraph product codes decoded using belief-propagation and ordered-statistics decoding [Panteleev and Kalachev, 2021]. We find that decoding data qubit and syndrome measurement errors together in a single stage leads to single-shot thresholds that greatly exceed all previously observed single-shot thresholds for these codes. For the 3D toric code and a phenomenological noise model, our results are consistent with a sustainable threshold of 7.1% for $Z$ errors, compared to the threshold of 2.90% previously found using a two-stage decoder~[Quintavalle et al., 2021]. For the 4D toric code, for which both $X$ and $Z$ error correction is single-shot, our results are consistent with a sustainable single-shot threshold of 4.3% which is even higher than the threshold of 2.93% for the 2D toric code for the same noise model but using $L$ rounds of stabiliser measurement. We also explore the performance of balanced product and 4D hypergraph product codes which we show lead to a reduction in qubit overhead compared the surface code for phenomenological error rates as high as 1%.

Overview of Improved Single-Shot Decoding of Higher Dimensional Hypergraph Product Codes

This paper presents a detailed investigation into the single-shot decoding of higher-dimensional hypergraph product codes utilizing belief propagation combined with ordered-statistics decoding (BP+OSD). The research focuses on the advantageous outcomes of decoding both data qubit and syndrome measurement errors in a unified step, consequently achieving single-shot thresholds that surpass previously established benchmarks for these types of codes.

Key Findings

• 3D and 4D Toric Code Performance: The paper reports a remarkable threshold of 7.1% for $Z$ errors in the 3D toric code under a phenomenological noise model, significantly improving upon prior results which identified a threshold of 2.90% using a different two-stage decoder approach. For the 4D toric code, where both $X$ and $Z$ error corrections are single-shot, a sustainable threshold of 4.3% is achieved. This represents an improvement compared to the 2D toric code's 2.93% threshold under similar conditions with $L$ rounds of stabilizer measurement.
• Hypergraph Product Code Efficiency: The research highlights that both balanced product and 4D hypergraph product codes can reduce qubit overhead compared to traditional surface codes, which becomes particularly noticeable at phenomenological error rates up to 1%.

Theoretical Implications

This study offers significant theoretical implications in quantum error correction, particularly regarding the use of hypergraph product codes for efficient quantum computation. The paper introduces advanced techniques in decoding strategies that encompass both data and measurement errors concurrently, offering significant improvements over classical methods that traditionally employ separate stages.

• Metacode Utilization in Chain Complexes: The research exploits the properties of chain complexes and $\mathbb{F}_2$-homology, which are instrumental when analyzing and constructing CSS codes. This methodological framework allows for more effective handling of measurement errors, a challenge in implementing fault-tolerant quantum computing systems.
• BP+OSD Integration: The integration of belief propagation with ordered-statistics decoding illustrates enhanced robustness against split-belief phenomena—a prevalent issue when decoding degenerately structured quantum error states. The utilization of metachecks in the Tanner graph structure further optimizes decoding accuracy and efficiency.

Practical Implications

On the practical side, the improved decoding strategies showcased in this research hold significant promise for the development of scalable quantum computing architectures by reducing computational overhead and enhancing error correction capabilities, particularly in noisy environments.

• Qubit Overhead Reduction: By demonstrating that balanced product codes achieve comparable error rates to large block size surface codes while utilizing significantly fewer qubits, this work hints at possible reductions in necessary physical resources for quantum error correction, contributing to more feasible quantum computing implementations.

Future Developments

Reflecting on the trajectory of advancements in this field, future research might explore further integration of BP+OSD within various classes of quantum codes, particularly focusing on mitigating computational overhead to facilitate real-time implementations. Expanding these decoding approaches to encompass circuit-level noise models could also yield deeper insights into practical applications in quantum hardware.

This paper's innovative methodologies and promising results represent a pivotal step toward advancing quantum error correction, impacting both theoretical frameworks and practical implementations of quantum computing technologies.