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A Systematic Approach to Hyperbolic Quantum Error Correction Codes

Ahmed Adel Mahmoud, Kamal Mohamed Ali, Steven Rayan

TL;DR

The paper addresses the design and analysis of hyperbolic quantum error correction codes (HQECCs) by leveraging hyperbolic tessellations and Fuchsian groups to realize qubit layouts on closed genus-$g$ Riemann surfaces. It introduces a systematic framework, anchored by a Hyperbolic Cycle Basis, to enumerate all plaquettes and nontrivial cycles, and integrates finite-$\{p,q\}$ lattices with PBCs to enable rigorous computation of code parameters. The work provides algorithms for constructing finite hyperbolic lattices, generating the cycle basis, and computing code distances, and demonstrates HQECCs based on $\{8,3\}$ and $\{10,3\}$ with estimated error thresholds of roughly $2$–$4\%$ and $4$–$6\%$, respectively, under depolarizing noise with a matching decoder. Together, these contributions establish a practical, scalable path toward systematic HQECC analysis and optimization in hyperbolic geometries, paving the way for robust quantum error correction beyond Euclidean codes.

Abstract

Hyperbolic quantum error correction codes (HQECCs) leverage the unique geometric properties of hyperbolic space to enhance the capabilities and performance of quantum error correction. By embedding qubits in hyperbolic lattices, HQECCs achieve higher encoding rates and improved error thresholds compared to conventional Euclidean codes. Building on recent advances in hyperbolic crystallography, we present a systematic framework for constructing HQECCs. As a key component of this framework, we develop a novel algorithm for computing all plaquette cycles and logical operators associated with a given HQECC. To demonstrate the effectiveness of this approach, we utilize this framework to simulate two HQECCs based respectively on two relevant examples of hyperbolic tilings. In the process, we evaluate key code parameters such as encoding rate, error threshold, and code distance for different sub-lattices. This work establishes a solid foundation for a systematic and comprehensive analysis of HQECCs, paving the way for the practical implementation of HQECCs in the pursuit of robust quantum error correction strategies.

A Systematic Approach to Hyperbolic Quantum Error Correction Codes

TL;DR

The paper addresses the design and analysis of hyperbolic quantum error correction codes (HQECCs) by leveraging hyperbolic tessellations and Fuchsian groups to realize qubit layouts on closed genus- Riemann surfaces. It introduces a systematic framework, anchored by a Hyperbolic Cycle Basis, to enumerate all plaquettes and nontrivial cycles, and integrates finite- lattices with PBCs to enable rigorous computation of code parameters. The work provides algorithms for constructing finite hyperbolic lattices, generating the cycle basis, and computing code distances, and demonstrates HQECCs based on and with estimated error thresholds of roughly – and –, respectively, under depolarizing noise with a matching decoder. Together, these contributions establish a practical, scalable path toward systematic HQECC analysis and optimization in hyperbolic geometries, paving the way for robust quantum error correction beyond Euclidean codes.

Abstract

Hyperbolic quantum error correction codes (HQECCs) leverage the unique geometric properties of hyperbolic space to enhance the capabilities and performance of quantum error correction. By embedding qubits in hyperbolic lattices, HQECCs achieve higher encoding rates and improved error thresholds compared to conventional Euclidean codes. Building on recent advances in hyperbolic crystallography, we present a systematic framework for constructing HQECCs. As a key component of this framework, we develop a novel algorithm for computing all plaquette cycles and logical operators associated with a given HQECC. To demonstrate the effectiveness of this approach, we utilize this framework to simulate two HQECCs based respectively on two relevant examples of hyperbolic tilings. In the process, we evaluate key code parameters such as encoding rate, error threshold, and code distance for different sub-lattices. This work establishes a solid foundation for a systematic and comprehensive analysis of HQECCs, paving the way for the practical implementation of HQECCs in the pursuit of robust quantum error correction strategies.

Paper Structure

This paper contains 7 sections, 2 theorems, 52 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Hyperbolic Geometry
  4. Fuchsian Groups
  5. Finite Hyperbolic lattices
  6. Hyperbolic Quantum Error Correction Codes
  7. Hyperbolic Cycle Basis

Key Result

Theorem 1

(Poincaré) A compact polygon $P$ satisfying the side and angle conditions is the fundamental domain of the group $\Gamma$ generated by the side-pairing transformations of $P$.

Figures (3)

  • Figure 1: The top half of the figure shows the unit cell of the $\{8,8\}$ Bravais lattice that, when compactified, can be embedded in the genus-$2$ Riemann surface shown in the bottom half of figure.
  • Figure 2: The top half shows finite {8,3} lattice generated by $N=9$ faces. The graph is generated by replicating the unit cell 8 times by applying the generators $\gamma_1,...,\gamma_4$ and their inverses. The bottom half shows the same lattice after imposing the PBCs.
  • Figure 3: Error threshold graph for the {8,3} HQECC.The error threshold is approximately in the range 2%-4%.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • proof