Generalized $\mathbb{Z}_p$ toric codes as qudit low-density parity-check codes
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866908848890052608 |
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| author | Liang, Zijian Chen, Yu-An |
| author_facet | Liang, Zijian Chen, Yu-An |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20158 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized $\mathbb{Z}_p$ toric codes as qudit low-density parity-check codes Liang, Zijian Chen, Yu-An Quantum Physics Strongly Correlated Electrons Mathematical Physics Quantum Algebra 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. |
| title | Generalized $\mathbb{Z}_p$ toric codes as qudit low-density parity-check codes |
| topic | Quantum Physics Strongly Correlated Electrons Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2602.20158 |