Algebraic Geometry Codes and Decoded Quantum Interferometry
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909830970605568 |
|---|---|
| author | Gu, Andi Jordan, Stephen P. |
| author_facet | Gu, Andi Jordan, Stephen P. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06603 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algebraic Geometry Codes and Decoded Quantum Interferometry Gu, Andi Jordan, Stephen P. Quantum Physics 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. |
| title | Algebraic Geometry Codes and Decoded Quantum Interferometry |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2510.06603 |