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Construction of a Family of Quantum Codes Using Sub-exceding Functions via the Hypergraph Product and the Generalized Shor Construction

Luc Rabefihavanana, Harinaivo Andriatahiny, Randriamiarampanahy Ferdinand

TL;DR

A new family of stabilizer quantum LDPC codes derived from the classical linear codes $L_k$ and $L_k^{+}$, defined via sub-exceding functions are introduced, establishing a new framework for structured quantum LDPC code design and optimization.

Abstract

In this paper, we introduce a new family of stabilizer quantum LDPC codes derived from the classical linear codes $L_k$ and $L_k^{+}$, defined via sub-exceding functions. In previous work, these codes demonstrated strong performance in minimum distance, decoding efficiency, and structural simplicity. By combining the hypergraph product framework with a generalized Shor construction, we obtain a scalable class of quantum codes with parameters $[[6k^2,\, k^2,\, d]]$. The resulting quantum codes exhibit a rich combinatorial structure and promising properties, particularly in terms of locality, low-density parity-check (LDPC) structure, and asymptotic behavior. The minimum distance satisfies $d=3$ for $k=3$ and $d=4$ for $k\ge4$, establishing a new framework for structured quantum LDPC code design and optimization.

Construction of a Family of Quantum Codes Using Sub-exceding Functions via the Hypergraph Product and the Generalized Shor Construction

TL;DR

A new family of stabilizer quantum LDPC codes derived from the classical linear codes and , defined via sub-exceding functions are introduced, establishing a new framework for structured quantum LDPC code design and optimization.

Abstract

In this paper, we introduce a new family of stabilizer quantum LDPC codes derived from the classical linear codes and , defined via sub-exceding functions. In previous work, these codes demonstrated strong performance in minimum distance, decoding efficiency, and structural simplicity. By combining the hypergraph product framework with a generalized Shor construction, we obtain a scalable class of quantum codes with parameters . The resulting quantum codes exhibit a rich combinatorial structure and promising properties, particularly in terms of locality, low-density parity-check (LDPC) structure, and asymptotic behavior. The minimum distance satisfies for and for , establishing a new framework for structured quantum LDPC code design and optimization.
Paper Structure (24 sections, 2 theorems, 56 equations)

This paper contains 24 sections, 2 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Presentation of the codes $L_k$ and $L_k^+$
  3. Sub-exceeding functions
  4. The linear code $L_k$
  5. The linear code $L_k^+$
  6. Explicit examples for the two codes $L_k$ and $L_k^{+}$
  7. Fundamentals and Structure of Stabilizer Codes
  8. Some quantum notions
  9. Pauli groups and stabilizer codes
  10. Symplectic product and construction of a stabilizer code from two classical binary codes
  11. Quantum code construction via the hypergraph product method
  12. Generalized Shor construction method
  13. Quantum code obtained from the two codes $\mathcal{L}_k$ and $\mathcal{L}^{+}_k$
  14. Rich combinatorial structure
  15. Minimum distance and error robustness
  16. ...and 9 more sections

Key Result

Theorem 1

A $[[n,k,d]]$ quantum stabilizer code $\mathcal{C}(\mathcal{S})$ is a subspace of $\mathcal{H}^{\otimes n}$ such that:

Theorems & Definitions (15)

  • Definition 1: Hilbert space of a qubit Tapp1999, Gilyen2019, Steane1996,Bodin2024
  • Definition 2: Tensor space of $n$ qubits Tapp1999, Steane1996, Baboin2011,Bodin2024
  • Definition 3: Quantum code Steane1996,Bodin2024, Calderbank1997
  • Definition 4
  • Definition 5: Pauli group for $n$ qubits Tapp1999,Steane1996,Tillich2008,Delfosse2012
  • Definition 6: Stabilizer group Steane1998, Calderbank1997, Lai2011, Cohen1998
  • Definition 7
  • Theorem 1: Parameters of stabilizer codes Lai2011, Cohen1998 , Delfosse2012
  • Definition 8: Symplectic representation Lai2011 , Delfosse2012
  • Example 1
  • ...and 5 more