Deflating quantum error-correcting codes

TL;DR

The paper introduces deflation as a generalized operation on quantum stabilizer codes that unifies shortening and puncturing, enabling finer control when removing multiple qudits. It formalizes deflation via the symplectic representation, analyzes parameter changes for pure and impure codes, and proves that deflation can increase dimension while reducing length, with minimum distance bounded by d-t. The authors show that deflation offers more flexibility than performing puncturing and shortening sequentially and provide an example demonstrating potentially better parameters. They also discuss the extension to classical codes and outline criteria and strategies to achieve improved distances, while acknowledging open questions on existence and optimal constructions.

Abstract

In this work, we introduce a technique for reducing the length of a quantum stabilizer code, and we call this deflation of the code. Deflation can be seen as a generalization of the well-known puncturing and shortening techniques in cases where more than a single qudit is removed. We show that the parameters of the deflated quantum code can be controlled, and argue that a similar approach is not as beneficial when applied to classical linear codes. Furthermore, it is shown that deflation introduces additional freedom compared to applying just puncturing and shortening consecutively. We exemplify that it is possible to obtain better parameters by deflating a code rather than consecutively using puncturing and shortening.

Theorems & Definitions (9)

• Proposition 1
• Proposition 2
• Lemma 1
• Theorem 1
• Lemma 2
• Lemma 3
• Theorem 2
• Example 1
• Proposition 3