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Synchronizable hybrid subsystem codes

Theerapat Tansuwannont, Andrew Nemec

TL;DR

This work addresses block synchronization errors in quantum communication by embedding quantum synchronizable codes (QSCs) into a broader CSS-based framework that also supports subsystem, hybrid, and hybrid subsystem codes. It shows how the same pair of classical cyclic codes with $C^\\perp \\subset C \\subset D$ yields a family of synchronizable codes and CSS-type constructions, enabling simultaneous protection against Pauli errors and misalignment while transmitting quantum and classical information. The main contributions include a unified family of synchronizable hybrid subsystem codes, a detailed encoding/decoding scheme, and explicit trade-offs between synchronization distance, gauge qubits, and encoded classical bits (e.g., $r+m+d_{sync,max}=2(k_d-k_c)$ for synchronizable codes). The paper also provides generalized methods to construct subsystem, hybrid, and hybrid subsystem codes of CSS type from classical codes beyond cyclic ones, and it discusses practical implications for modular quantum architectures and potential extensions to qudit systems.

Abstract

Quantum synchronizable codes are quantum error correcting codes that can correct not only Pauli errors but also errors in block synchronization. The code can be constructed from two classical cyclic codes $\mathcal{C}$, $\mathcal{D}$ satisfying $\mathcal{C}^{\perp} \subset \mathcal{C} \subset \mathcal{D}$ through the Calderbank-Shor-Steane (CSS) code construction. In this work, we establish connections between quantum synchronizable codes, subsystem codes, and hybrid codes constructed from the same pair of classical cyclic codes. We also propose a method to construct a synchronizable hybrid subsystem code which can correct both Pauli and synchronization errors, is resilient to gauge errors by virtue of the subsystem structure, and can transmit both classical and quantum information, all at the same time. The trade-offs between the number of synchronization errors that the code can correct, the number of gauge qubits, and the number of logical classical bits of the code are also established. In addition, we propose general methods to construct hybrid and hybrid subsystem codes of CSS type from classical codes, which cover relevant codes from our main construction.

Synchronizable hybrid subsystem codes

TL;DR

This work addresses block synchronization errors in quantum communication by embedding quantum synchronizable codes (QSCs) into a broader CSS-based framework that also supports subsystem, hybrid, and hybrid subsystem codes. It shows how the same pair of classical cyclic codes with yields a family of synchronizable codes and CSS-type constructions, enabling simultaneous protection against Pauli errors and misalignment while transmitting quantum and classical information. The main contributions include a unified family of synchronizable hybrid subsystem codes, a detailed encoding/decoding scheme, and explicit trade-offs between synchronization distance, gauge qubits, and encoded classical bits (e.g., for synchronizable codes). The paper also provides generalized methods to construct subsystem, hybrid, and hybrid subsystem codes of CSS type from classical codes beyond cyclic ones, and it discusses practical implications for modular quantum architectures and potential extensions to qudit systems.

Abstract

Quantum synchronizable codes are quantum error correcting codes that can correct not only Pauli errors but also errors in block synchronization. The code can be constructed from two classical cyclic codes , satisfying through the Calderbank-Shor-Steane (CSS) code construction. In this work, we establish connections between quantum synchronizable codes, subsystem codes, and hybrid codes constructed from the same pair of classical cyclic codes. We also propose a method to construct a synchronizable hybrid subsystem code which can correct both Pauli and synchronization errors, is resilient to gauge errors by virtue of the subsystem structure, and can transmit both classical and quantum information, all at the same time. The trade-offs between the number of synchronization errors that the code can correct, the number of gauge qubits, and the number of logical classical bits of the code are also established. In addition, we propose general methods to construct hybrid and hybrid subsystem codes of CSS type from classical codes, which cover relevant codes from our main construction.
Paper Structure (27 sections, 16 theorems, 76 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 27 sections, 16 theorems, 76 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical cyclic codes
  4. Stabilizer codes and the CSS construction
  5. Subsystem codes, hybrid codes, and hybrid subsystem codes
  6. Subsystem codes
  7. Hybrid codes
  8. Hybrid subsystem codes
  9. Quantum synchronizable codes in stabilizer formalism
  10. Encoding procedure
  11. Correcting bit-flip, phase-flip, and synchronization errors
  12. From generators of classical cyclic codes to anticommuting pairs of Pauli operators
  13. A family of synchronizable hybrid subsystem codes
  14. Subsystem and synchronizable subsystem codes
  15. Hybrid and synchronizable hybrid codes
  16. ...and 12 more sections

Key Result

Theorem 1

CS96Steane96b Let $\mathcal{C}_x$ and $\mathcal{C}_z$ be classical linear codes with parameters $\left[n,k_x\right]$ and $\left[n,k_z\right]$ respectively. Let $\mathcal{C}_z^\perp\subseteq \mathcal{C}_x$. Then there exists a quantum stabilizer code with stabilizer group $\left\langle X(\mathcal{C}_

Figures (3)

  • Figure 1: A circuit diagram describing the entire encoding procedure of a QSC.
  • Figure 2: A Venn diagram displaying properties of the codes in a family of the synchronizable hybrid subsystem codes from \ref{['thm:QSC_unified']}.
  • Figure 3: Relationship between each of the codes from \ref{['thm:QSC_unified']} and their initial codes. The constructions of these codes are related by three key processes: the encoding procedure involving ancilla attachment and CNOT operations, gauge fixing, and the sacrifice of synchronization distance.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Lemma 1
  • proof
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • ...and 15 more