Constructing Quantum CSS Codes with Improved Parameters through Lifts

Core Concepts

This article introduces a general method to lift quantum CSS codes by constructing covering spaces of their associated Tanner cone-complex. The lift can be applied to various quantum CSS code constructions, including hypergraph product codes, and enables the generation of new families of quantum CSS codes with potentially improved parameters.

Abstract

The article proposes a notion of lift for quantum CSS codes, inspired by the geometrical construction of Freedman and Hastings. It is based on the existence of a canonical complex associated to any CSS code, called the Tanner cone-complex, over which covering spaces are generated.
As a first application, the authors describe the classification of lifts of hypergraph product codes (HPC) and demonstrate the equivalence with the lifted product code (LPC) construction. This shows that the lift of a CSS code can be seen as a generalization of the LPC.
As a second application, the authors report several new non-product constructions of quantum CSS codes and apply the lift prescription to generate their lifts, which for certain covering maps, are codes with improved relative parameters compared to the initial one.
The parameters, dimension and distance, of a lifted CSS code are in general hard to determine. The authors analyze the case of lifting an arbitrary HPC and show that the lift is more general than simply lifting the two linear codes, factors of the HPC, independently. They provide a complete classification of lifts of HPCs based on the notion of Goursat quintuples.
The lift of a quantum CSS code can also be expressed as a left or right module over a certain group ring, when the associated covering is regular. This constitutes a systematic way of generating balanced product of quantum CSS codes, as first pointed out by Breuckmann and Eberhardt.
Finally, the authors introduce three new families of quantum CSS codes specifically designed to be lifted. For each of them, the associated Tanner cone-complex has a fundamental group isomorphic to a predetermined infinite group, enabling the generation of lifted codes with potentially improved parameters.

Stats

The length n of the lifted CSS code is an integer multiple of the length of the input code.
The maximum weight of rows and columns of the lifted check matrices is unchanged compared to that of the input code.

Quotes

"The key ingredient to define the lift of a quantum CSS code is a geometrical object faithfully representing the code, similarly to the fact that a lift of a linear code can be obtained from a covering of its Tanner graph."
"The lift of a quantum CSS code enjoys the following properties: 1) For an input CSS code of length n, the lift is a code of size given by an integer multiple of n, 2) The maximum weight of rows and columns of the lifted check matrices is unchanged compared to that of the input code, 3) Applied to a classical code, it coincides with the lift of linear codes, 4) Applied to an HPC, it coincides with the LPC construction described in [PK22a]."

Key Insights Distilled From

Lifts of quantum CSS codes

by Virgile Guem... at arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16736.pdf

Deeper Inquiries

How can the parameters, dimension and distance, of a lifted CSS code be rigorously analyzed and optimized

To rigorously analyze and optimize the parameters of a lifted CSS code, several steps can be taken:

Mathematical Formulation: Start by defining the lift operation in a precise mathematical framework. Clearly define how the lift affects the code parameters, such as dimension and distance.

Parameter Analysis: Analyze the impact of the lift on the parameters of the CSS code. Understand how the dimension and distance change with different types of lifts and covering maps.

Case Studies: Conduct case studies with specific examples of CSS codes and their lifts. Calculate the parameters for each case and observe any patterns or trends that emerge.

Optimization Techniques: Explore optimization techniques to improve the parameters of the lifted codes. This could involve finding the best covering maps or voltage assignments that lead to enhanced code properties.

Comparative Analysis: Compare the parameters of the lifted CSS codes with those of the original codes. Identify scenarios where the lift operation results in significant improvements in dimension or distance.

Theoretical Bounds: Consider theoretical bounds on the parameters of quantum codes and see if the lifted CSS codes approach or surpass these bounds. This can provide insights into the optimality of the lifted codes.

By following these steps and conducting a thorough analysis, it is possible to rigorously analyze and optimize the parameters of lifted CSS codes.

What are the limitations and potential drawbacks of the lift construction compared to other quantum CSS code design methods

The lift construction for quantum CSS codes, while powerful and versatile, has certain limitations and potential drawbacks compared to other quantum CSS code design methods:

Complexity: The lift operation can be complex, especially when dealing with large codes or intricate covering maps. This complexity can make the analysis and optimization of parameters challenging.

Parameter Control: The control over the parameters of the lifted CSS codes may not be as straightforward as with other design methods. It may require a detailed understanding of the underlying mathematical structures.

Resource Intensive: Implementing the lift operation for quantum CSS codes may require significant computational resources, especially for codes with large lengths or high-dimensional spaces.

Dependence on Covering Maps: The effectiveness of the lift construction heavily relies on the choice of covering maps. Finding optimal covering maps for desired code properties can be a non-trivial task.

Generalization: The lift operation is specifically tailored for CSS codes, and generalizing it to other types of quantum error correcting codes may not be straightforward. Different code families may require different lifting techniques.

While the lift construction offers unique advantages in generating new quantum CSS codes, it is essential to consider these limitations and drawbacks when utilizing this method for code design.

Can the lift operation be generalized to other types of quantum error correcting codes beyond CSS codes

The lift operation can potentially be generalized to other types of quantum error correcting codes beyond CSS codes, but it would require careful consideration and adaptation to the specific characteristics of those codes. Here are some key points to consider:

Code Structure: Different quantum error correcting codes have distinct structures and properties. The lift operation would need to be modified to accommodate the specific requirements of each code family.

Parameter Preservation: When generalizing the lift to other code types, it is crucial to ensure that important parameters, such as dimension and distance, are preserved or optimized through the lifting process.

Covering Maps: The choice of covering maps plays a significant role in the success of the lift operation. For different types of codes, suitable covering maps need to be identified to ensure the effectiveness of the lift.

Mathematical Framework: Developing a generalized lift operation for various quantum error correcting codes would involve establishing a robust mathematical framework that can be applied across different code families.

Experimental Validation: Before applying the lift to other code types, it would be beneficial to experimentally validate its effectiveness and impact on the parameters of the resulting codes.

By carefully adapting the lift operation to suit the characteristics of different quantum error correcting codes and conducting thorough analyses, it may be possible to generalize the lift to a broader range of code families.