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Graded Quantum Codes

Tony Shaska

Abstract

This work develops a geometric framework for constructing quantum error-correcting codes from weighted projective and orbifold structures, integrating algebraic geometry, divisor theory, and the CSS stabilizer formalism. Beginning with weighted projective spaces and their associated height and defect structures, the study builds classical AG-codes via evaluation on divisors adapted to orbifold singularities. These classical codes are lifted to quantum codes using self-orthogonality conditions and homological constructions, yielding a class of Quantum Weighted Algebraic Geometric (QWAG) codes. A central contribution is the formulation of a refined Singleton-type bound motivated by orbifold defect terms and effective genus corrections. While the classical quantum Singleton bound is recovered in the smooth case, the orbifold setting suggests additional geometric contributions that may adjust the theoretical distance bound. The refined bound is presented with partial justification under specific geometric hypotheses and framed as a conjectural extension in full generality. The monograph further provides explicit constructions, computational implementations in Sage/Python, and illustrative examples demonstrating how weighted geometry influences code parameters. This work establishes a structured bridge between orbifold geometry and quantum coding theory, outlining both concrete constructions and open problems for further mathematical development.

Graded Quantum Codes

Abstract

This work develops a geometric framework for constructing quantum error-correcting codes from weighted projective and orbifold structures, integrating algebraic geometry, divisor theory, and the CSS stabilizer formalism. Beginning with weighted projective spaces and their associated height and defect structures, the study builds classical AG-codes via evaluation on divisors adapted to orbifold singularities. These classical codes are lifted to quantum codes using self-orthogonality conditions and homological constructions, yielding a class of Quantum Weighted Algebraic Geometric (QWAG) codes. A central contribution is the formulation of a refined Singleton-type bound motivated by orbifold defect terms and effective genus corrections. While the classical quantum Singleton bound is recovered in the smooth case, the orbifold setting suggests additional geometric contributions that may adjust the theoretical distance bound. The refined bound is presented with partial justification under specific geometric hypotheses and framed as a conjectural extension in full generality. The monograph further provides explicit constructions, computational implementations in Sage/Python, and illustrative examples demonstrating how weighted geometry influences code parameters. This work establishes a structured bridge between orbifold geometry and quantum coding theory, outlining both concrete constructions and open problems for further mathematical development.

Paper Structure

This paper contains 159 sections, 77 theorems, 355 equations, 7 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite fields
  4. Existence and Uniqueness
  5. Properties
  6. Polynomial Rings
  7. Linear Codes
  8. Encoding, Generator, and Parity-Check Matrices
  9. Syndrome and Error Detection
  10. Weight Distribution and MacWilliams Identities
  11. Fundamental Bounds
  12. Classical Examples
  13. From Linear to Algebraic Geometry Codes
  14. AG codes on curves
  15. Introduction to Algebraic Curves over Finite Fields
  16. ...and 144 more sections

Key Result

Proposition 2.1

For any prime $p$ and integer $r \geq 1$, there exists a finite field with $p^r$ elements lidl1997finite.

Figures (7)

  • Figure 1: Mixed states correspond to interior points of the Bloch sphere, with the center representing maximal uncertainty.
  • Figure 2: Classical NOT versus quantum $X$ gate acting on basis states.
  • Figure 3: Hadamard: creates superposition from $\lvert 0 \rangle$; applying again interferes back to $\lvert 0 \rangle$.
  • Figure 4: CNOT: flips target iff control is $\lvert 1 \rangle$ (XOR-like behavior).
  • Figure 5: Circuit creating the Bell state $\tfrac{1}{\sqrt{2}}(\lvert 00 \rangle+\lvert 11 \rangle)$.
  • ...and 2 more figures

Theorems & Definitions (245)

  • Proposition 2.1: Existence of Finite Fields
  • Example 2.2: Finite Field $\mathbb{F}_4$
  • Proposition 2.3: Singleton Bound
  • Proposition 2.4: Hamming or Sphere-Packing Bound
  • Proposition 2.5: Gilbert–Varshamov Bound
  • Example 2.6: Repetition Code
  • Example 2.7: Hamming Code
  • Example 2.8: Cyclic and BCH Codes
  • Example 2.9: Reed–Solomon Code
  • Theorem 2.10: Hasse–Weil Bound
  • ...and 235 more