Galois hull dimension improves quantum error-correcting code design
Galois Hulls of Generalized Roth-Lempel Codes and Their Applications to EAQECCs
Information Theory
Summary
Quantum computers need special codes to protect information from errors. These codes often rely on mathematical objects called Galois hulls. The authors studied a type of code called generalized Roth-Lempel codes and found ways to build them with precise control over the Galois hull’s size. This control allows creating better quantum error-correcting codes that use entanglement efficiently and have good error protection. Their methods provide explicit constructions useful for developing such quantum codes.
What this means in practice
- •For quantum hardware engineers: Create quantum error-correcting codes that consume known amounts of entanglement while maintaining high error tolerance in quantum processors.
- •For quantum communication developers: Design quantum communication protocols using entanglement-assisted codes with customizable hull dimensions to optimize error resistance and entanglement usage.
Authors
Xuefei Wu, Qi Liu, Yingchun Chen, Haiyan Zhou
Abstract
The dimension of a Galois hull is an important parameter in the construction of entanglement-assisted quantum error-correcting codes. In this paper, we study generalized Roth-Lempel (GRL) codes with prescribed Galois hull dimensions over finite fields of arbitrary characteristic. We develop a common construction method based on normalized Lagrange coefficients and suitable choices of column multipliers, and obtain six explicit constructions from multiplicative cosets, trace fibers, additive-subspace cosets, and mixed additive--multiplicative fibers. For extension sizes $s=2$ and $s=3$, we construct MDS and AMDS GRL codes with Galois hulls of specific dimensions. The resulting codes provide EAQECCs with explicit dimension, entanglement consumption, and relatively large minimum distance. Taking zero-dimensional hulls also gives Hermitian LCD GRL codes and the corresponding maximally entangled EAQECCs.