The Impact of Metric Selection and Algorithmic Optimisation on Large-Scale Surface Codes in Quantum Error Correction

Mehmet Keçeci

ORCID : https://orcid.org/0000-0001-9937-9839

Received: 03.10.2025

“Article 8 of the series”

Abstract:

This study, based on “Metric Selection in Quantum Error Correction” [Unpublished pre-doctoral VIII. technical reports, Gebze Technical University, Kocaeli, Türkiye (312, 466, 474, 475)], focuses on enhancing the performance of Quantum Error Correction (QEC) codes, which play a critical role in overcoming decoherence and operational errors—among the most significant obstacles to practical quantum computing. It provides a comprehensive investigation into the effects of metric selection and algorithmic optimisations on the error correction efficiency of large-scale surface codes (toric codes). A key finding is that the type of metric used to define the distance between physical qubits (Euclidean, Minkowski, Manhattan, and potentially Riemannian) significantly impacts both the execution time and accuracy of QEC decoders. Simulations involving high qubit counts (up to 250,000) demonstrated that while the Euclidean metric generally offers a good balance, the Minkowski metric provides flexibility through its adjustable p-values. Despite the potential for higher accuracy with a Riemannian metric, challenges in integration with existing decoders and its computational cost currently limit its practical applicability. Within the context of algorithmic optimisations, the performance of the Minimum Weight Perfect Matching (MWPM) and the Union-Find (UF) algorithms was compared. At high qubit counts, MWPM was generally observed to yield superior results. To further enhance the efficiency of these decoders, the open-source Blossom V library was recompiled and optimised using modern, high-performance languages such as C++ (employing C++20/C++23 standards) and Rust. These compilation optimisations yielded remarkable speed-ups of up to ~190x in solution times for large-scale systems, particularly in trials using the g++ compiler. The work also discusses, on a theoretical level, the specific error types encountered in advanced qubit designs, such as cat-qubits (notably phase-flip errors), and how these might be addressed by leveraging fundamental physical principles like the Aharonov-Bohm effect. In conclusion, this research underscores the critical importance of a synergistic approach combining metric selection, algorithm design, and software optimisation for the development of large-scale, fault-tolerant quantum computers. The findings are poised to inform the future shaping of QEC strategies and quantum hardware-software co-design.

Keywords: Quantum Error Correction, Surface Codes, Toric Code, Metric Selection, Euclidean, Minkowski, Minimum Weight Perfect Matching, MWPM, Blossom V, Algorithmic Optimisation, C++ Optimisation, Large-Scale Simulations, Cat State, Cat-Qubit Errors, Union-Find, UF.

Note: Citations and numbering are in continuation of the previous articles.