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Algebraic Geometry Codes and Decoded Quantum Interferometry

Andi Gu, Stephen P. Jordan

TL;DR

The paper extends Decoded Quantum Interferometry (DQI) beyond Reed–Solomon codes by formulating Hermitian Optimal Polynomial Intersection (HOPI) to operate with Hermitian codes, enabling larger block lengths $n=q^3$ over $\mathbb{F}_{q^2}$ and reducing qubit overhead per field element. It leverages the dual Hermitian-code structure and existing decoding algorithms to show that DQI retains a quantum advantage in finding high-quality approximate optima for algebraic-geometry–based regression problems, as quantified by the semicircle law. A precise performance formula ties the expected fraction of satisfied constraints to the dual-distance parameters and decoding radius, and empirical comparisons against Prange’s information-set decoding demonstrate scalable quantum advantage, including regimes where the advantage grows with problem size and peaks near $r/q^2 \approx 0.28$. These results indicate the DQI framework’s advantage is not restricted to RS codes and suggest a broader applicability to structured optimization problems on algebraic varieties, guiding future exploration of additional AG-code families and their practical quantum implementations.

Abstract

Decoded Quantum Interferometry (DQI) defines a duality that pairs decoding problems with optimization problems. The original work on DQI considered Reed-Solomon decoding, whose dual optimization problem, called Optimal Polynomial Intersection (OPI), is a polynomial regression problem over a finite field. Here, we consider a class of algebraic geometry codes called Hermitian codes, which achieve block length $q^3$ using alphabet $\mathbb{F}_{q^2}$ compared to Reed-Solomon's limitation to block length $q$ over $\mathbb{F}_q$, requiring approximately one-third fewer qubits per field element for quantum implementations. We show that the dual optimization problem, which we call Hermitian Optimal Polynomial Intersection (HOPI), is a polynomial regression problem over a Hermitian curve, and because the dual to a Hermitian code is another Hermitian code, the HOPI problem can also be viewed as approximate list recovery for Hermitian codes. By comparing to Prange's algorithm, simulated annealing, and algebraic list recovery algorithms, we find a large parameter regime in which DQI efficiently achieves a better approximation than these classical algorithms, suggesting that the apparent quantum speedup offered by DQI extends beyond Reed-Solomon codes to a broader class of polynomial regression problems on algebraic varieties.

Algebraic Geometry Codes and Decoded Quantum Interferometry

TL;DR

The paper extends Decoded Quantum Interferometry (DQI) beyond Reed–Solomon codes by formulating Hermitian Optimal Polynomial Intersection (HOPI) to operate with Hermitian codes, enabling larger block lengths over and reducing qubit overhead per field element. It leverages the dual Hermitian-code structure and existing decoding algorithms to show that DQI retains a quantum advantage in finding high-quality approximate optima for algebraic-geometry–based regression problems, as quantified by the semicircle law. A precise performance formula ties the expected fraction of satisfied constraints to the dual-distance parameters and decoding radius, and empirical comparisons against Prange’s information-set decoding demonstrate scalable quantum advantage, including regimes where the advantage grows with problem size and peaks near . These results indicate the DQI framework’s advantage is not restricted to RS codes and suggest a broader applicability to structured optimization problems on algebraic varieties, guiding future exploration of additional AG-code families and their practical quantum implementations.

Abstract

Decoded Quantum Interferometry (DQI) defines a duality that pairs decoding problems with optimization problems. The original work on DQI considered Reed-Solomon decoding, whose dual optimization problem, called Optimal Polynomial Intersection (OPI), is a polynomial regression problem over a finite field. Here, we consider a class of algebraic geometry codes called Hermitian codes, which achieve block length using alphabet compared to Reed-Solomon's limitation to block length over , requiring approximately one-third fewer qubits per field element for quantum implementations. We show that the dual optimization problem, which we call Hermitian Optimal Polynomial Intersection (HOPI), is a polynomial regression problem over a Hermitian curve, and because the dual to a Hermitian code is another Hermitian code, the HOPI problem can also be viewed as approximate list recovery for Hermitian codes. By comparing to Prange's algorithm, simulated annealing, and algebraic list recovery algorithms, we find a large parameter regime in which DQI efficiently achieves a better approximation than these classical algorithms, suggesting that the apparent quantum speedup offered by DQI extends beyond Reed-Solomon codes to a broader class of polynomial regression problems on algebraic varieties.

Paper Structure

This paper contains 26 sections, 6 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background
  4. Notation
  5. Decoded Quantum Interferometry
  6. Hermitian Codes
  7. Dual Codes.
  8. Decoding Algorithms.
  9. Hermitian Optimal Polynomial Intersection
  10. Results
  11. Limits and Open Questions
  12. Basics of Algebraic Geometry over Finite Fields
  13. Algebraic Varieties and Curves
  14. Affine and Projective Space
  15. Algebraic Varieties
  16. ...and 11 more sections

Key Result

Theorem 1

Let $q$ be a prime power and consider an instance of HOPI over the Hermitian curve with $n=q^3$ constraints, alphabet $\mathbb{F}_{q^2}$, and constraint sets of size $r$. Let $\mathcal{C}^\perp$ be the dual Hermitian code of minimum distance $d^\perp$, and assume efficient decoding up to $\ell = \lf

Figures (2)

  • Figure 1: Performance of DQI on HOPI. Panel (a) shows the balanced case ($r=q^2/2$) for $q^2=25$ ($n=125$), plotting the expected satisfaction fraction $\expval{s}/n$ against code rate $k/n$. DQI follows the semicircle curve, while Prange's performance increases linearly. Panel (b) fixes $k/n=0.2$ and shows performance as $q$ grows, confirming that the quantum advantage persists at large $n$.
  • Figure 2: Advantage ratio $\expval{s}_{\mathrm{DQI}}/\expval{s}_{\mathrm{Prange}}$ as a function of $n$ and $r$. The rate is fixed at $k/n=0.2$. The black line marks the constraint-set size $r$ at which the ratio is maximized, occurring around $r/q^2\approx 0.28$.

Theorems & Definitions (18)

  • Theorem 1: DQI performance for HOPI
  • Definition 1: Affine space
  • Definition 2: Projective space
  • Definition 3: Affine variety
  • Definition 4: Irreducible variety
  • Definition 5: Smooth curve
  • Definition 6: Function field
  • Definition 7: Divisor
  • Definition 8: Principal divisor
  • Theorem 2: Degree zero property
  • ...and 8 more