[Paper Review] Entanglement-Assisted and Subsystem Quantum Codes: New Propagation Rules and Constructions

Gaojun Luo, Martianus Frederic Ezerman+1 more|arXiv (Cornell University)|Jun 20, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces new propagation rules for entanglement-assisted and subsystem quantum error-correcting codes (EAQECCs and subsystem codes) by analyzing the Hermitian hulls of punctured and shortened generalized Reed-Solomon (GRS) codes. By constructing $k$-dimensional GRS codes with $(k-1)$-dimensional Hermitian hulls that are maximum distance separable (MDS), the authors derive new families of optimal EAQECCs and subsystem codes with improved parameters, demonstrating that puncturing and shortening preserve desirable hull properties and yield optimal quantum codes under the Singleton-like bound.

ABSTRACT

This paper proposes new propagation rules on quantum codes in the entanglement-assisted and in quantum subsystem scenarios. The rules lead to new families of such quantum codes whose parameters are demonstrably optimal. To obtain the results, we devise tools to puncture and shorten codes in ways that ensure their Hermitian hulls have certain desirable properties. More specifically, we give a general framework to construct $k$-dimensional generalized Reed-Solomon codes whose Hermitian hulls are $(k-1)$-dimensional maximum distance separable codes.

Motivation & Objective

• To develop new propagation rules for entanglement-assisted and subsystem quantum codes that extend beyond the stabilizer framework.
• To address the lack of systematic classical code constructions that yield optimal quantum codes in the EAQECC and subsystem frameworks.
• To identify classical linear codes—specifically GRS codes—whose Hermitian hulls have desirable MDS properties to enable optimal quantum code constructions.
• To demonstrate that puncturing and shortening operations preserve optimal hull structures, enabling parameter optimization in quantum codes.
• To provide a general framework for constructing optimal EAQECCs and subsystem codes with parameters matching or approaching the Singleton-like bound.

Proposed method

• The authors analyze the Hermitian hulls of generalized Reed-Solomon (GRS) codes and derive conditions under which the hull is an MDS code of dimension $k-1$ for a $k$-dimensional GRS code.
• They introduce a framework to puncture and shorten GRS codes in a way that maintains the desired Hermitian hull structure, ensuring the resulting classical codes retain optimal parameters.
• The propagation rules are derived by applying Lemma 1 from [6], which links the parameters of the EAQECC to the dimension of the Hermitian hull and the dual code of the classical ingredient.
• The method leverages the fact that puncturing preserves the hull dimension under specific conditions, enabling the construction of new EAQECCs with $c$ entangled pairs.
• The approach is generalized to subsystem codes by applying similar puncturing and shortening techniques to classical codes with controlled hulls.
• Theoretical results are supported by explicit constructions and tables listing parameters of known and new optimal EAQECCs and subsystem codes for reference.

Experimental results

Research questions

• RQ1 Can new propagation rules be established for entanglement-assisted and subsystem quantum codes that go beyond the stabilizer framework and preserve optimality?
• RQ2 What conditions on classical linear codes—specifically GRS codes—ensure their Hermitian hulls are MDS and of dimension $k-1$?
• RQ3 How can puncturing and shortening operations be applied to classical codes to preserve optimal hull structures and yield optimal quantum codes?
• RQ4 Do the resulting EAQECCs and subsystem codes achieve the Singleton-like bound, and can their parameters be systematically derived from classical code parameters?
• RQ5 Can the framework be extended to non-MDS classical codes or other inner products (e.g., Euclidean) to yield new quantum code families?

Key findings

• The authors construct $k$-dimensional GRS codes over $\mathbb{F}_{q^2}$ whose Hermitian hulls are $(k-1)$-dimensional MDS codes, providing a key ingredient for optimal quantum code constructions.
• By puncturing and shortening such codes, they derive new families of $[[n, \kappa, \delta; c]]_q$ EAQECCs with parameters that are demonstrably optimal under the Singleton-like bound.
• The propagation rules allow the construction of $[[n-s, \kappa, \geq \delta; s]]_q$ EAQECCs from a pure $[[n, \kappa, \delta]]_q$ code, extending known results to a broader class of codes.
• For subsystem codes, the authors present new parameters in Table III, with most entries being optimal and not previously reported in the literature.
• The construction yields optimal subsystem codes with lengths exceeding $q+1$, including cases where $n = q^2 - 1 - s$ and $n = q^2 - i(q+1) - s$, achieving $\delta = k - s + 1$ and $r = k - s$.
• The results confirm that the proposed method produces quantum codes that meet or approach the Singleton-like upper bounds, with explicit parameter tables provided for $q \in \{2,3\}$ to support practical implementation.