Quantum codes from one-point codes on norm-trace curves

Gretchen L. Matthews, Julia Shapiro

In 2025, Jordan et al. introduced Decoded Quantum Interferometry (DQI), a quantum algorithm for combinatorial optimization based on decoding. They considered Reed-Solomon decoding and its associated optimization problem, called Optimal Polynomial Intersection (OPI), which may be viewed as a polynomial regression problem over a finite field. DQI exhibits provable speedups on certain problem instances and establishes a connection between decoding problems and optimization tasks. Leveraging the well-understood dual structure and decoding theory of algebraic geometry codes from other curve families, we introduce COFI: Curve-based Optimal Function Intersection. By considering two-point Hermitian codes, Suzuki codes, and extended norm-trace codes, we broaden the range of algebraic geometry codes used in DQI and identify families that offer further improvements over one-point Hermitian codes in the Hermitian Optimal Polynomial Intersection framework considered by Jordan and Gu. Depending on the family and parameter regime, these curves can reduce quantum resource requirements or increase the number of constraints that can be considered.

Pavan Kumar, Noor Mohammad Khan

Carlos Galindo, Fernando Hernando, Diego Ruano

Martianus Frederic Ezerman, Radoslav Kirov

Sarvepalli and Klappenecker showed how classical one-point codes on the Hermitian curve can be used to construct quantum codes. Homma and Kim determined the parameters of a larger family of codes, the two-point codes. In quantum error-correction, the observed presence of asymmetry in some quantum channels led to the study of asymmetric quantum codes (AQECCs) where we no longer assume that the different types of errors are equiprobable. This paper considers quantum codes constructed from the two-point codes. In the asymmetric case, we show strict improvements over all possible finite fields for a range of designed distances. We produce large dimension pure AQECC and small dimension impure AQECC that have better parameters than AQECC from one-point codes. Numerical results for the Hermitian curves over F16 and F64 are used to illustrate the gain.

Lin Sok, San Ling

This article studies the Hermitian hulls of linear codes from algebraic plane curves of a special type. We provide lower bounds on the hull dimension of one-point algebraic geometry codes. We derive two families of entanglement-assisted quantum codes from one-point algebraic geometry codes, using the Hermitian approach.

Giuliano G. La Guardia