Graded Quantum Codes

Abstract

This thesis introduces Quantum Weighted Algebraic Geometry Codes (QWAGs), a novel class of quantum error-correcting codes derived from weighted superelliptic curves over finite fields. By extending classical algebraic geometry codes, the weighted framework incorporates graded rings and orbifold corrections, enhancing parameter flexibility and self-orthogonality for Calderbank–Shor–Steane (CSS) constructions. We develop divisor theory, Riemann–Roch spaces, and duality in quasi-smooth weighted settings, proving Euclidean duality via canonical divisors and residue pairings. These foundations enable QWAC families with parameters shaped by graded geometry. A homological perspective through evaluation chain complexes elucidates CSS conditions, leading to a refined quantum Singleton bound incorporating orbifold terms. Complementing the theory, we present a Python-based computational framework automating curve construction, point enumeration, Riemann–Roch computations, and CSS verification. This work unifies weighted algebraic geometry, coding theory, and quantum stabilizers, advancing tools for quantum error correction.