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On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes

Carlos Galindo, Fernando Hernando

TL;DR

The paper generalizes the standard Hermitian self-orthogonal code construction for stabilizer quantum codes to extension fields $\mathbb{F}_{q^{2m}}$, showing that any Hermitian self-orthogonal $[n,k]_{q^{2m}}$ code yields an $[[mn, mn-2mk, \ge d^{\perp_h}]]_q$ stabilizer code. It provides a practical Jin-based method to generate such Hermitian self-orthogonal codes and a $K_n$-based bound to ensure broad applicability. The authors produce a wealth of new codes, including numerous binary records and expansive non-binary families over fields such as $\mathbb{F}_4, \mathbb{F}_7, \mathbb{F}_8, \mathbb{F}_9$, often surpassing existing results (SLLG, Cao-Cui). Overall, the work expands the catalog of quantum codes with long lengths and improved parameters, potentially enhancing fault-tolerant quantum computing capabilities.

Abstract

Many $q$-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal $q^2$-ary linear codes. This result can be generalized to $q^{2 m}$-ary linear codes, $m > 1$. We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to \cite{codet} and new $q$-ary ones, with $q \neq 2$, improving others in the literature.

On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes

TL;DR

The paper generalizes the standard Hermitian self-orthogonal code construction for stabilizer quantum codes to extension fields , showing that any Hermitian self-orthogonal code yields an stabilizer code. It provides a practical Jin-based method to generate such Hermitian self-orthogonal codes and a -based bound to ensure broad applicability. The authors produce a wealth of new codes, including numerous binary records and expansive non-binary families over fields such as , often surpassing existing results (SLLG, Cao-Cui). Overall, the work expands the catalog of quantum codes with long lengths and improved parameters, potentially enhancing fault-tolerant quantum computing capabilities.

Abstract

Many -ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal -ary linear codes. This result can be generalized to -ary linear codes, . We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to \cite{codet} and new -ary ones, with , improving others in the literature.

Paper Structure

This paper contains 6 sections, 4 theorems, 13 equations, 3 tables.

Table of Contents

  1. A construction of stabilizer quantum codes
  2. Hermitian self-orthogonal codes and examples
  3. A useful result
  4. Examples
  5. Binary stabilizer quantum codes
  6. Non-binary stabilizer quantum codes

Key Result

Theorem 1.1

Let $C$ be an $\mathbb{F}_{q^2}$-linear code of length $n$ and dimension $k$. Assume that $C$ is Hermitian self-orthogonal, i.e. Then, there exists a stabilizer quantum code over $\mathbb{F}_{q}$ with parameters $[[n,n-2k, \geq d^{\perp_h}]]_q$, where $d^{\perp_h}$ stands for the minimum distance of the code $C^{\perp_h}$.

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3