On geometric aspects of quantum stabilizer codes

Quantum computers promise to solve problems much faster than regular computers, from fields of secure communication to medicine. Unlike classical computers, which process information using binary bits (0s and 1s), quantum computers use qubits, which can exist in multiple states at the same time. This property allows them to perform complex calculations exponentially faster than traditional systems. However, quantum computers face a big challenge: errors. Qubits are very sensitive and can be easily disturbed by their environments. When errors happen, they can affect calculations, making it difficult for quantum computers to work correctly. So how can we protect quantum information from these disturbances? Scientists develop quantum error correction codes that help detect and fix these errors to overcome these difficulties. Among these codes, quantum stabilizer codes play a key role in ensuring reliable quantum computing. While the mathematical foundations of these codes are well established, their connection to geometry remains less explored. This project aims to study quantum stabilizer codes from a geometric perspective. We will investigate how mathematical structures, such as points and lines in finite spaces, can help improve these codes. By translating quantum coding problems into geometric language, we hope to discover new and better ways to protect quantum information. One of our main goals is to explore whether certain types of quantum stabilizer codes exist, and if so, how they can be constructed. We will also examine how these codes behave over different types of number systems, particularly those with odd characteristics, which have not been widely studied. Additionally, we will investigate a special group of error-correcting codes known as maximum distance separable (MDS) codes to see if they can be adapted to quantum computing. By combining knowledge from quantum physics, algebra, and geometry, this research takes an innovative approach to solving a key challenge in quantum computing. The results could contribute to making quantum computers more reliable and practical for real-world applications.