Improving Error Correction Capability of CSS Codes Using Symbol-Pair Decoding

Concepts de base

The error correctability of CSS codes obtained from classical cyclic codes can be improved by lifting the syndrome decoder for codes over the symbol-pair metric.

Résumé

The paper presents a novel approach to improve the error correction capability of CSS (Calderbank-Shor-Steane) quantum error-correcting codes by leveraging the relationship between the symplectic weight and the symbol-pair weight.

Key highlights:

1. The authors establish a connection between the symplectic weight and the symbol-pair weight, which allows them to lift decoders for classical codes over the symbol-pair metric to decoders for stabilizer quantum codes.
2. They propose a new decoding algorithm for CSS codes derived from classical cyclic codes satisfying the dual-containing property (C⊥Euc ⊆ C). The algorithm uses the syndrome decoding of codes over the symbol-pair metric to correct errors with weight up to ⌊(dp-1)/2⌋ in the symplectic weight and ⌊(dH-1)/2⌋ in the Hamming weight.
3. The authors show that their decoding scheme can correct a larger set of errors compared to the previous decoding schemes for CSS codes, as the symbol-pair distance of cyclic codes is at least 3/2 times their Hamming distance.
4. The paper provides an illustrative example demonstrating the improvement in error correction capability using the proposed decoding scheme.

Stats

The symbol-pair weight of a vector x := (x0, x1, ..., xn-1) ∈ Fn2 is defined as:
wtsp(x) := |{i : (xi, xi+1) ≠ (0, 0)}|
The symplectic weight of a vector (a|b) := ((a0, ..., an-1)|(b0, ..., bn-1)) ∈ (Fn2)2 is defined as:
wtsymp(a|b) := |{1 ≤ i ≤ n : (ai, bi) ≠ (0, 0)}|
For a binary cyclic code C, the following relation holds:
dp(C) ≥ 3dH(C)/2

Citations

"The relation between stabilizer codes and binary codes provided by Gottesman and Calderbank et al. is a celebrated result, as it allows the lifting of classical codes to quantum codes."
"An equivalent way to state this result is that the work allows us to lift decoders for classical codes over the Hamming metric to decoders for stabilizer quantum codes."
"A natural question to consider: Can we do something similar with decoders for classical codes considered over other metrics? i.e., Can we lift decoders for classical codes over other metrics to obtain decoders for stabilizer quantum codes?"

Idées clés tirées de

A Symbol-Pair Decoder for CSS Codes

by Vatsal Pramo... à arxiv.org 09-18-2024

https://arxiv.org/pdf/2409.10979.pdf

Questions plus approfondies

How can the proposed decoding scheme be extended to more general families of stabilizer codes beyond CSS codes?

The proposed decoding scheme, which improves the error correctability of CSS codes by utilizing the symbol-pair metric, can potentially be extended to more general families of stabilizer codes by exploring the underlying structure of these codes. One approach is to analyze the properties of the parity-check matrices associated with various stabilizer codes, not just CSS codes.
For instance, the relationship between the symbol-pair metric and the symplectic weight can be generalized to other types of stabilizer codes by identifying how the error correction capabilities of these codes can be enhanced through similar syndrome decoding techniques. By establishing a framework that connects the symbol-pair metric with the specific characteristics of other stabilizer codes, such as those derived from different classical codes or those that do not adhere to the dual-containing property, we can adapt the decoding algorithms accordingly.
Moreover, the extension could involve investigating the use of more complex error models that account for different types of errors beyond bit-flip and phase-flip errors. This would require a deeper understanding of the algebraic structures of the stabilizer codes and how they interact with the symbol-pair metric. By leveraging the insights gained from CSS codes, researchers can develop new decoding strategies that are applicable to a broader class of stabilizer codes, potentially leading to improved performance in quantum error correction.

Can the connection between the symplectic weight and the symbol-pair weight be leveraged to obtain improved bounds or constructions for quantum error-correcting codes?

Yes, the connection between the symplectic weight and the symbol-pair weight can indeed be leveraged to obtain improved bounds and constructions for quantum error-correcting codes. The established relationship, as indicated in the paper, shows that the symbol-pair weight can provide a more nuanced understanding of the error-correcting capabilities of quantum codes derived from classical codes.
By utilizing the inequalities that relate the symbol-pair weight to the Hamming weight, researchers can derive tighter bounds on the minimum distance of quantum codes. For example, the inequality (wtH(x) + 1 \leq wtsp(x) \leq 2wtH(x)) suggests that the symbol-pair metric can yield a more favorable error-correcting performance compared to the traditional Hamming metric. This insight can lead to the construction of quantum codes that are capable of correcting a greater number of errors than previously thought, thereby enhancing their practical utility.
Furthermore, the relationship can be used to explore new families of quantum codes that are optimized for specific applications, such as high-density data storage systems where the symbol-pair metric is particularly relevant. By focusing on the symplectic weight in conjunction with the symbol-pair weight, researchers can develop innovative coding strategies that push the boundaries of what is achievable in quantum error correction.

What other applications or insights can be derived from the relationship between classical codes over the symbol-pair metric and quantum stabilizer codes?

The relationship between classical codes over the symbol-pair metric and quantum stabilizer codes opens up several avenues for applications and insights in the field of quantum information theory.

Enhanced Error Correction in Quantum Systems: The insights gained from this relationship can lead to the development of more robust quantum error-correcting codes that are specifically designed to handle errors arising in practical quantum computing scenarios, such as those encountered in high-density storage systems. By applying the symbol-pair metric, researchers can create codes that are better suited for environments where adjacent symbol errors are prevalent.

Interdisciplinary Applications: The principles derived from the symbol-pair metric can be applied to other areas of information theory and coding, such as classical error correction in communication systems. This cross-pollination of ideas can lead to improved coding strategies in both classical and quantum domains, enhancing overall data integrity and reliability.

New Constructions of Quantum Codes: The relationship can inspire new constructions of quantum codes that leverage the unique properties of the symbol-pair metric. By exploring how classical codes can be transformed into quantum codes through this lens, researchers may discover novel coding schemes that outperform existing ones in terms of error correction capabilities.

Theoretical Insights into Quantum Mechanics: Understanding the interplay between classical coding theory and quantum error correction can provide deeper theoretical insights into the foundations of quantum mechanics. This could lead to advancements in quantum cryptography and secure communication protocols, where error correction plays a critical role in maintaining the integrity of quantum information.

In summary, the relationship between classical codes over the symbol-pair metric and quantum stabilizer codes not only enhances our understanding of quantum error correction but also paves the way for innovative applications and theoretical advancements across multiple domains.