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

• URL: http://arxiv.org/abs/2602.20158v1
• Date: Mon, 23 Feb 2026 18:59:31 GMT
• Title: Generalized $\mathbb{Z}_p$ toric codes as qudit low-density parity-check codes
• Abstract summary: We study two-dimensional translation-invariant CSS stabilizer codes over prime-dimensional qudits on the square lattice under twisted boundary conditions.<n>We find that the best observed $k d2$ at $n$ increases with $p$, with an empirical relation $k d2 = 0.0541, n2ln p + 3.84, n$, compatible with a Bravyi--Poulin--Terhal-type tradeoff when the interaction range grows with system size.
• Score: 5.692499671837265
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 Gröbner 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.

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