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

Figures (4)

• Figure 1: (a) The $A_v$ and $B_p$ stabilizers of the generalized ${\mathbb Z}_p$ toric codes, parameterized by the Laurent polynomials $f(x,y)=1+r_1 x+r_2 x^{a}y^{b}$ and $g(x,y)=1+r_3 y+r_4 x^{c}y^{d}$ in Eq. \ref{['eq: stabilizer']}, with $r_1,r_2,r_3,r_4\in{\mathbb Z}_p \setminus\{0\}$ and $a,b,c,d\in{\mathbb Z}$. The green-shaded edges denote the unit cell at the origin used to generate the Pauli module over the Laurent polynomial ring haah_module_13. For the same pair $(f,g)$, different choices of twisted boundary conditions can realize distinct quantum LDPC codes. (b) Adapted from Ref. liang2025Generalized. A twisted torus embedded in three-dimensional space. The twist is applied along the longitudinal cycle by an angle that is a fraction of $2\pi$, as indicated by the red curve tracing a noncontractible cycle. The twisted torus is specified by two vectors $\vec{a}_1=(0,\alpha)$ and $\vec{a}_2=(\beta,\gamma)$, i.e., lattice sites are identified by $\vec{v}\sim \vec{v}+\vec{a}_1\sim \vec{v}+\vec{a}_2$ for all $\vec{v}$.
• Figure 2: Representative stabilizer realizations on twisted tori. (a) $X$- and $Z$-stabilizers of the $[[242,10,22]]_3$ code with $f=1-x+x^3y^{-3}$ and $g=1-y+x^{-4}$, defined with twisted boundary vectors $\vec{a}_1=(0,11)$ and $\vec{a}_2=(11,4)$. (b) $X$- and $Z$-stabilizers of the $[[120,6,20]]_{11}$ code with $f=1+x+2x^{-2}y^{3}$ and $g=1+y+4x^{-1}y^{-4}$, defined with $\vec{a}_1=(0,10)$ and $\vec{a}_2=(6,2)$. All other stabilizers are obtained by translations. In each panel, vertices with the same letter (top/bottom) or the same number (left/right) are identified, yielding a twisted torus.
• Figure 3: Finite-size performance across qudit dimensions. (a) $[[n,k,d]]_p$ LDPC codes from Tables \ref{['tab: n_k_d 1']}--\ref{['tab: n_k_d 5']}: the metric $k d^{2}/n$ is plotted versus block length $n$ for $p\in\{2,3,5,7,11\}$, together with linear fits for each $p$. (b) Slopes extracted from the fits in panel (a), plotted against $\ln(p)$, with error bars given by the standard errors from the linear fits in panel (a). Linear regression indicates an approximately linear dependence on $\ln(p)$.
• Figure 4: Finite-size performance metric $k d^{2}/n$ for all code instances listed in Tables \ref{['tab: n_k_d 1']}–\ref{['tab: n_k_d 5']}, plotted against $(\ln p)\,n$ for the explored primes $p\in{2,3,5,7,11}$. The solid line is the least-squares linear fit $k d^{2}/n = 0.0541\,(\ln p)\,n + 3.84$ with $R^{2}=0.96$.