Graded Quantum Codes: From Weighted Algebraic Geometry to Homological Chain Complexes

Abstract: We introduce graded quantum codes, unifying two classes of quantum error-correcting codes. The first, quantum weighted algebraic geometry (AG) codes, derives from rational points on hypersurfaces in weighted projective spaces over finite fields. This extends classical AG codes by adding weighted degrees and singularities, enabling self-orthogonal codes via the CSS method with improved distances using algebraic structures and invariants like weighted heights.The second class arises from chain complexes of graded vector spaces, generalizing homological quantum codes to include torsion and multiple gradings. This produces low-density parity-check codes with parameters based on homology ranks, including examples from knot invariants and quantum rotors. A shared grading leads to a refined Singleton bound: $d \leq \frac{n - k + 2}{2} - \frac{\epsilon}{2}$, where $\epsilon > 0$ reflects entropy adjustments from geometric singularities and defects. The bound holds partially for simple orbifolds and is supported by examples over small fields. Applications include post-quantum cryptography, fault-tolerant quantum computing, and optimization via graded neural networks, linking algebraic geometry, homological algebra, and quantum information.