Generalized $\mathbb{Z}_p$ toric codes as qudit low-density parity-check codes

TL;DR

This paper generalizes the $ ext{Z}_p$ toric code to qudit low-density parity-check codes, using algebraic methods to efficiently analyze their properties and identify optimal finite-size codes with high performance.

Contribution

It introduces a systematic approach to construct and analyze $ ext{Z}_p$ toric codes as qudit LDPC codes, extending the Kitaev model and employing Laurent-polynomial formalism with Gr"obner bases.

Findings

Identified specific qudit LDPC codes with high $k d^{2}/n$ ratios.

Established an empirical relation between $k d^{2}$, system size, and prime dimension $p$.

Demonstrated the scalability and performance tradeoffs of these codes.

Abstract

We study two-dimensional translation-invariant CSS stabilizer codes over prime-dimensional qudits on the square lattice under twisted boundary conditions, generalizing the Kitaev $\mathbb{Z}_p$ toric code by augmenting each stabilizer with two additional qudits. Using the Laurent-polynomial formalism, we adapt the Gr\"obner basis to compute the logical dimension $k$ efficiently, without explicitly constructing large parity-check matrices. We then perform a systematic search over various stabilizer realizations and lattice geometries for $p\in\{3,5,7,11\}$, identifying qudit low-density parity-check codes with the optimal finite-size performance. Representative examples include $[[242,10,22]]_3$ and $[[120,6,20]]_{11}$, both achieving $k d^{2}/n=20$. Across the searched regime, the best observed $k d^{2}$ at fixed $n$ increases with $p$, with an empirical relation $k d^{2} = 0.0541 \, n^{2}\ln p + 3.84 \, n$, compatible with a Bravyi--Poulin--Terhal-type tradeoff when the interaction range grows with system size.