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Constructions and decoding procedures for quantum CSS codes

Yannick Saouter, Massinissa Zenia, Gilles Burel

TL;DR

The article develops new quantum CSS codes by applying Sloane's classical self-orthogonal constructions to quantum settings, enabling families with explicit distance $d=\min(d_1,d_2)$ and structured stabilizer form. It introduces an algebraic decoding approach that leverages decoders for the duals $C_1^{\perp}$ or $C_2^{\perp}$ to perform CSS decoding, reducing reliance on large lookup tables. Concrete instantiations are provided using BCH, Reed-Muller, and finite projective geometry codes, with Rudolph's majority-vote decoding enabling algebraic recovery for the duals. Extensive enumerations up to length $128$ illustrate the practicality of the constructions, including several optimal or near-optimal CSS codes and the potential to generalize to broader stabilizer codes.

Abstract

This article presents new constructions of quantum error correcting Calderbank-Shor-Steane (CSS for short) codes. These codes are mainly obtained by Sloane's classical combinations of linear codes applied here to the case of self-orthogonal linear codes. A new algebraic decoding technique is also introduced. This technique is exemplified on CSS codes obtained from BCH, Reed-Muller and projective geometry codes.

Constructions and decoding procedures for quantum CSS codes

TL;DR

The article develops new quantum CSS codes by applying Sloane's classical self-orthogonal constructions to quantum settings, enabling families with explicit distance and structured stabilizer form. It introduces an algebraic decoding approach that leverages decoders for the duals or to perform CSS decoding, reducing reliance on large lookup tables. Concrete instantiations are provided using BCH, Reed-Muller, and finite projective geometry codes, with Rudolph's majority-vote decoding enabling algebraic recovery for the duals. Extensive enumerations up to length illustrate the practicality of the constructions, including several optimal or near-optimal CSS codes and the potential to generalize to broader stabilizer codes.

Abstract

This article presents new constructions of quantum error correcting Calderbank-Shor-Steane (CSS for short) codes. These codes are mainly obtained by Sloane's classical combinations of linear codes applied here to the case of self-orthogonal linear codes. A new algebraic decoding technique is also introduced. This technique is exemplified on CSS codes obtained from BCH, Reed-Muller and projective geometry codes.

Paper Structure

This paper contains 24 sections, 21 theorems, 30 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Quantum transmission channel
  3. Stabilizer and CSS codes
  4. New self-orthogonal codes from old ones
  5. Proof :
  6. Proof :
  7. Proof :
  8. Proof :
  9. Proof :
  10. Proof :
  11. Proof :
  12. Proof :
  13. Proof :
  14. Proof :
  15. Proof :
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\mathcal{C}$ be a stabilizer code of $n$ qubits stabilized by $\mathcal{S}$. Let $r$ be the number of independent generators of $\mathcal{S}$. We have then:

Figures (3)

  • Figure 1: Encoding procedure of $\mathsf{C_1}\bigstar_{\varphi} \mathsf{C_2}$
  • Figure 2: Decoding procedure for CSS quantum codes
  • Figure 3: Projective geometry $PG(2,2)$ (Fano triangle)

Theorems & Definitions (43)

  • Definition 1: Qubit
  • Definition 2: Quantum state
  • Definition 3: Pauli group
  • Definition 4
  • Definition 5: Pauli communication channel
  • Definition 6: Depolarizing channel
  • Definition 7: Stabilizer code
  • Definition 8: Dimension of a stabilizer code
  • Theorem 1
  • Definition 9: Centralizer
  • ...and 33 more