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Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma

Michael Liaofan Liu, Nathanan Tantivasadakarn, Victor V. Albert

TL;DR

This work develops a linear-algebraic, symplectic framework for subsystem stabilizer codes and their CSS generalizations. It introduces a doubling map $\Delta$ that converts any modular-qudit subsystem stabilizer code into a subsystem CSS code with parameters $[[2n,2k,2r,d']]$ satisfying $d \le d' \le 2d$, and provides a Steane-type decoder using only the two underlying classical codes. A Goursat-Lemma based decomposition shows every subsystem stabilizer code can be built from two nested subsystem CSS codes, enabling a structured understanding and new code families. The results unify quantum-subsystem and classical-code perspectives, preserve locality in the mappings, and yield practical recovery procedures, while suggesting directions for fault-tolerant implementations and extensions to broader quantum-code frameworks.

Abstract

The CSS code construction is a powerful framework used to express features of a quantum code in terms of a pair of underlying classical codes. Its subsystem extension allows for similar expressions, but the general case has not been fully explored. Extending previous work of Aly, Klappenecker, and Sarvepalli [quantph/0610153], we determine subsystem CSS code parameters, express codewords, and develop a Steane-type decoder using only data from the two underlying classical codes. Generalizing a result of Kovalev and Pryadko [Phys. Rev. A 88 012311 (2013)], we show that any subsystem stabilizer code can be "doubled" to yield a subsystem CSS code with twice the number of physical, logical, and gauge qudits and up to twice the code distance. This mapping preserves locality and is tighter than the Majorana-based mapping of Bravyi, Terhal, and Leemhuis [New J. Phys. 12 083039 (2010)]. Using Goursat's Lemma, we show that every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints, and we characterize subsystem stabilizer codes based on the nested codes' properties.

Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma

TL;DR

This work develops a linear-algebraic, symplectic framework for subsystem stabilizer codes and their CSS generalizations. It introduces a doubling map that converts any modular-qudit subsystem stabilizer code into a subsystem CSS code with parameters satisfying , and provides a Steane-type decoder using only the two underlying classical codes. A Goursat-Lemma based decomposition shows every subsystem stabilizer code can be built from two nested subsystem CSS codes, enabling a structured understanding and new code families. The results unify quantum-subsystem and classical-code perspectives, preserve locality in the mappings, and yield practical recovery procedures, while suggesting directions for fault-tolerant implementations and extensions to broader quantum-code frameworks.

Abstract

The CSS code construction is a powerful framework used to express features of a quantum code in terms of a pair of underlying classical codes. Its subsystem extension allows for similar expressions, but the general case has not been fully explored. Extending previous work of Aly, Klappenecker, and Sarvepalli [quantph/0610153], we determine subsystem CSS code parameters, express codewords, and develop a Steane-type decoder using only data from the two underlying classical codes. Generalizing a result of Kovalev and Pryadko [Phys. Rev. A 88 012311 (2013)], we show that any subsystem stabilizer code can be "doubled" to yield a subsystem CSS code with twice the number of physical, logical, and gauge qudits and up to twice the code distance. This mapping preserves locality and is tighter than the Majorana-based mapping of Bravyi, Terhal, and Leemhuis [New J. Phys. 12 083039 (2010)]. Using Goursat's Lemma, we show that every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints, and we characterize subsystem stabilizer codes based on the nested codes' properties.
Paper Structure (28 sections, 18 theorems, 181 equations, 2 figures, 1 table)

This paper contains 28 sections, 18 theorems, 181 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and summary of results
  2. Preliminaries
  3. Pauli groups
  4. Pauli groups as vector spaces
  5. Subsystem stabilizer codes
  6. Subsystem stabilizer codes as subspaces of vector spaces
  7. Subsystem CSS codes
  8. Stabilizer-to-CSS mapping
  9. Recovery procedure for subsystem CSS codes
  10. Quantum step: syndrome measurement
  11. Classical step: error recovery
  12. Structure of subsystem stabilizer codes
  13. Goursat data of the gauge group
  14. Goursat data of the stabilizer group
  15. Characterizing subsystem stabilizer codes using the Goursat data of the stabilizer group
  16. ...and 13 more sections

Key Result

Proposition 1

Let $\omega$ be an antisymmetric form on $G\times G$, let $H\leq G\times G$, and let $n \coloneqq \dim G$. Then \begin{tikzpicture} [ node distance = \nd and \nd, on grid, baseline = (current bounding box.center) ] \node (1) {\(G \times G\)}; \node (2) [below=of 1] {\(H for some $d \in \mathbb{Z}$.

Figures (2)

  • Figure 1: Illustration of Lemma \ref{['lem:Goursat']} and Proposition \ref{['prop:GoursatProperties']} with $|{E_X}/{N_X}|= 4 =|{E_Z}/{N_Z}|$. Any subspace $H \leq G \times G$ consists of "blocks" --- i.e., cosets --- (dark shaded rectangles) of a direct product ${N_X}\times {N_Z}$ (bottom left dark shaded rectangle) contained inside a direct product ${E_X}\times {E_Z}$ (large rectangle). Shaded regions belong to $H$, while unshaded regions do not. Here, $\phi \left(v_X + {N_X}\right)= v_Z+{N_Z}$ for $v = s,t,u$.
  • Figure 2: Venn diagram depicting features of subsystem stabilizer codes with maximal and minimal stabilizer groups, per Definition \ref{['def:maxmincenter']}. A code has minimal stabilizer iff its stabilizer group is CSS. A code has minimal and maximal stabilizer iff its gauge group is CSS. Every subspace stabilizer code has maximal stabilizer, but there are codes with maximal stabilizer that are not subspace stabilizer codes.

Theorems & Definitions (44)

  • Proposition 1
  • proof
  • Definition 1
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Example 1: Five-qubit code
  • ...and 34 more