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Systematic approach to hyperbolic quantum error correction codes

Ahmed Adel Mahmoud1,2,*, Kamal Mohamed Ali1, and Steven Rayan1,2,†

  • *Contact author: ahmed.mahmoud@usask.ca
  • †Contact author: rayan@math.usask.ca
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Phys. Rev. A 113, 042426 – Published 10 April, 2026

DOI: https://doi.org/10.1103/95mp-w7kr

Abstract

Quantum error correction codes defined on hyperbolic lattices leverage the unique geometric properties of the hyperbolic space to enhance the performance of quantum error correction. By embedding qubits in hyperbolic lattices, these codes achieve higher encoding rates and lower qubit overhead compared to those defined on conventional Euclidean lattices. Building on recent advances in hyperbolic crystallography, we introduce a unified framework for the systematic construction and scalable benchmarking of CSS quantum error correction codes on hyperbolic lattices. A central component of this framework is the Hyperbolic Cycle Basis algorithm, which employs graph-theoretic methods to efficiently identify all plaquette cycles (parity-check supports) and nontrivial cycles (logical operators). This enables scalable and automated benchmarking of a broad class of CSS codes defined on hyperbolic geometries. We apply this framework to construct and simulate two representative hyperbolic quantum error correction codes (HQECCs), evaluating key performance metrics such as encoding rate, error threshold, and code distance for different sublattices. While HQECCs serve as concrete examples, the framework can be adapted to a wide range of CSS codes, including those with more intricate stabilizer structures such as Floquet codes. This work establishes a foundation for systematic exploration and benchmarking of CSS codes on hyperbolic lattices, paving the way toward practical, high-performance quantum error correction.

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Phys. Rev. A 113, 042426– Published 10 April, 2026
  • Received 12 November 2025
  • Accepted 13 March 2026
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