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

Generalized Z p toric co des as qudit lo w-densit y parit y-c hec k co des Zijian Liang 1 and Y u-An Chen 1 , ∗ 1 International Center for Quantum Materials, Scho ol of Physics, Peking University, Beijing 100871, China (Dated: F ebruary 24, 2026) W e study tw o-dimensional translation-in v arian t CSS stabilizer co des o v er prime-dimensional qu- dits on the square lattice under t wisted boundary conditions, generalizing the Kitaev Z p toric co de b y augmenting each stabilizer with tw o additional qudits. Using the Laurent-polynomial formalism, w e adapt the Gr¨ obner basis to compute the logical dimension k efficien tly , without explicitly con- structing large parit y-chec k matrices. W e then perform a systematic searc h ov er v arious stabilizer realizations and lattice geometries for p ∈ { 3 , 5 , 7 , 11 } , iden tifying qudit low-densit y parity-c heck co des with the optimal finite-size p erformance. Representativ e examples include [[242 , 10 , 22]] 3 and [[120 , 6 , 20]] 11 , b oth achieving kd 2 /n = 20. Across the searched regime, the b est 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 Bra vyi–Poulin–T erhal–t yp e tradeoff when the in teraction range gro ws with system size. In tro duction. — F ault-tolerant quantum computation requires activ e protection against noise, and quan- tum error-correcting codes are essen tial to ac hiev e this goal [ 1 – 5 ]. T op ological stabilizer codes, such as toric co des, are promising because they use lo cal parit y chec ks to ac hieve high thresholds on tw o-dimensional arc hitec- tures [ 6 – 18 ]. Recen t dev elopments on biv ariate bicy- cle (BB) codes sho w that t wo-dimensional translation- in v arian t CSS codes with weigh t-6 chec ks can outper- form the standard toric code on relativ ely small tori, with p erformance improv ed b y nearly an order of mag- nitude [ 19 – 32 ]. Most of the existing literature, ho wev er, fo cuses on qubit co des, ev en though several experimental platforms naturally realize higher-level systems suc h as qutrits or more general qudits [ 33 – 41 ]. This motiv ates the question of ho w to design and analyze efficient quan- tum low-densit y parit y-c hec k (LDPC) co des directly in the qudit setting [ 42 ]. A t an abstract level, tw o-dimensional translation- in v arian t Pauli stabilizer co des ov er prime-dimensional qudits that satisfy the top ological order (TO) condition are kno wn to b e locally equiv alent to stacks of Z p toric co des and trivial product states [ 43 – 51 ]. The language of topological order and top ological quan tum field the- ory (TQFT) [ 52 – 88 ], such as any on types, fusion and braiding, ground-state degeneracy , and Wilson-line op- erators, can therefore be imported directly in to the qudit stabilizer setting. F rom this p ersp ective, a Z p qudit sta- bilizer code can b e view ed as a discrete Z p top ological gauge theory equipp ed with a built-in notion of topolog- ical excitations and their syndromes. In parallel, an algebraic viewpoint base d on Laurent p olynomial rings provides a pow erful framew ork for an- alyzing translation-inv ariant stabilizer codes [ 31 , 46 , 47 , 89 ]. By represen ting P auli operators as mo dules o v er the p olynomial ring R = Z p [ x ± 1 , y ± 1 ], one enco des lo cal- it y and translation inv ariance in to polynomial data and extracts the asso ciated an yon conten t via ring-theoretic metho ds. Gr¨ obner-basis techniques then enable efficient ∗ E-mail: yuanchen@pku.edu.cn computation of the ground-state degeneracy and logi- cal op erators on twisted tori, without constructing large parit y-chec k matrices. In this work, we adapt this ma- c hinery to prime-dimensional qudits, focusing on gener- alized Z p toric co des in whic h the standard X -star and Z -plaquette stabilizers are augmented b y tw o additional qudits, yielding w eight-6 chec ks. W e then systematically searc h for lo w-ov erhead qudit LDPC co des at system sizes of a few h undred qudits. Summary of results. — W e study t wo-dimensional translation-in v arian t CSS co des ov er Z p qudits with p ∈ { 3 , 5 , 7 , 11 } , sp ecified b y a pair of Laurent p olynomials f ( x, y ) , g ( x, y ) ∈ R . This p olynomial data determines the stabilizers of generalized Z p toric co des on t wisted tori. W e systematically enumerate lo cal stabilizer real- izations together with twisted b oundary conditions, and compute the corresp onding code parameters [[ n, k, d ]] p , where n is the n umber of physical qudits, k is the n um- b er of logical qudits, and d is the code distance. F or p = 3 w e searc h system sizes up to n ≤ 250, while for p ∈ { 5 , 7 , 11 } w e search up to n ≤ 150. The resulting high-p erforming qudit LDPC co des and their [[ n, k , d ]] p parameters are summarized in T ables I – IV . W e explicitly present these [[ n, k , d ]] p co des via their stabilizers (Fig. 1(a) ) and the t wisted torus (Fig. 1(b) ). F or fixed parameters [[ n, k , d ]] p , there are t ypically man y c hoices of stabilizer polynomials and lattice v ectors; in the tables, we highlight realizations with the most lo cal stabilizers, whic h are most promising for implemen tation on qudit hardw are. Represen tative examples include [[52 , 8 , 7]] 3 , [[78 , 6 , 11]] 3 , [[156 , 10 , 15]] 3 , [[242 , 10 , 22]] 3 , [[30 , 4 , 7]] 5 , [[40 , 4 , 9]] 5 , [[78 , 10 , 9]] 5 , [[120 , 6 , 17]] 5 , [[28 , 4 , 7]] 7 , [[96 , 6 , 15]] 7 , [[98 , 10 , 12]] 7 , [[144 , 6 , 21]] 7 , [[40 , 4 , 10]] 11 , [[100 , 6 , 17]] 11 , [[120 , 6 , 20]] 11 , [[144 , 6 , 23]] 11 . In particular, the co des [[242 , 10 , 22]] 3 and [[120 , 6 , 20]] 11 b oth attain k d 2 /n = 20; their stabilizers and twisted tori are shown in Fig. 2 . Moreo ver, within the searc hed regime the b est observed kd 2 at fixed n increases with p , sho wn as Fig. 3 , and the fit in Fig. 4 yields the empirical relation k d 2 = 0 . 0541 n 2 ln p + 3 . 84 n . 2 (a) Generalized toric co des ov er Z p . (b) Twisted torus in three-dimensional space. FIG. 1. (a) The A v and B p stabilizers of the generalized Z p toric co des, parameterized by the Laurent p olynomials 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. ( 5 ), with r 1 , r 2 , r 3 , r 4 ∈ Z p \ { 0 } and a, b, c, d ∈ Z . The green-shaded edges denote the unit cell at the origin used to generate the P auli mo dule o v er the Lauren t p olynomial ring [ 46 ]. F or the same pair ( f , g ), different c hoices of twisted b oundary conditions can realize distinct quantum LDPC codes. (b) Adapted from Ref. [ 90 ]. A twisted torus embedded in three-dimensional space. The twist is applied along the longitudinal cycle b y an angle that is a fraction of 2 π , as indicated by the red curve tracing a noncontractible cycle. The t wisted torus is specified b y t w o v ectors  a 1 = (0 , α ) and  a 2 = ( β , γ ), i.e., lattice sites are identified by  v ∼  v +  a 1 ∼  v +  a 2 for all  v . Review of ring-theoretical approac h for biv ariate bicycle co des. — In this section, w e review the algebraic framew ork for analyzing translation-inv ariant quan tum co des on lattices [ 47 ]. W e follo w the notation of Ref. [ 90 ] and extend its Gr¨ obner-basis analysis to Z p qudits. W e b egin by recalling the standard p × p (generalized) Pauli op erators for a Z p qudit: X = X j ∈ Z p | j + 1 ⟩⟨ j | , Z = X j ∈ Z p ω j | j ⟩⟨ j | . (1) where ω is defined as ω := exp  2 π i d  . More explicitly , X =       0 0 · · · 0 1 1 0 · · · 0 0 0 1 · · · 0 0 . . . . . . . . . . . . . . . 0 0 · · · 1 0       , Z =        1 0 0 · · · 0 0 ω 0 · · · 0 0 0 ω 2 · · · 0 . . . . . . . . . . . . . . . 0 0 0 · · · ω d − 1        , (2) and X and Z satisfy the commutation relation Z X = ω X Z . (3) F or simplicit y , w e fo cus on the square lattice with one Z p qudit living at each edge. W e briefly review the p oly- nomial represen tation of Pauli op erators [ 46 ]. A unit cell con tains tw o vertices, e 1 and e 2 , whose P auli op erators are represented by four-dimensional vectors: X e 1 =       1 0 0 0       , Z e 1 =       0 0 1 0       , X e 2 =       0 1 0 0       , Z e 2 =       0 0 0 1       . (4) A translation b y ( n, m ) ∈ Z 2 is implemented by multi- plying the corresp onding v ector b y the monomial x n y m . More generally , an y Pauli operator (modulo an o v erall phase) is represented by a vector in R 4 , where R = Z p [ x ± 1 , y ± 1 ]. V ector addition corresponds to op erator m ultiplication, while m ultiplication b y elemen ts of R en- co des lattice translations and Z p exp onen ts. Thus, the P auli group mo dulo phase is naturally identified with an R -mo dule. A translation-inv ariant CSS code is sp ecified by a pair of Laurent p olynomials f , g ∈ R : A v =        f ( x, y ) g ( x, y ) 0 0        , B p =        0 0 − g ( x, y ) f ( x, y )        , (5) where ( · ) denotes the antipo de map x n y m 7→ x − n y − m . The stabilizer group is generated by all lattice translates of A v and B p . As a simple example, the Kitaev toric co de [ 6 ] corresp onds to f ( x, y ) = 1 − x and g ( x, y ) = 1 − y . The top ological order condition is satisfied when f ( x, y ) and g ( x, y ) are coprime [ 31 ], which ensures that an y lo cal operator commuting with all stabilizers is it- self a stabilizer. Moreov er, the maximal num ber of logi- cal qudits on a torus equals the num ber of indep enden t an yon generators in the underlying topological order [ 92 – 94 ]. It can b e computed from the co dimension of the ideal generated b y f and g : k max = 2 dim Å Z p [ x ± 1 , y ± 1 ] ⟨ f , g ⟩ ã . (6) F or twisted boundary conditions sp ecified by  a 1 = (0 , α ) and  a 2 = ( β , γ ), the resulting co de has logical dimen- sion [ 90 ] k = 2 dim Å Z p [ x ± 1 , y ± 1 ] ⟨ f ( x, y ) , g ( x, y ) , y α − 1 , x β y γ − 1 ⟩ ã . (7) 3 [[ n, k, d ]] 3 f ( x, y ) g ( x, y )  a 1  a 2 kd 2 n [[16 , 4 , 4]] 3 1 + x + y 1 + y − x − 1 y (0 , 4) (2 , 1) 4 [[26 , 6 , 5]] 3 1 + x + x − 1 y 1 − y − x − 1 (0 , 13) (1 , 5) 5.77 [[42 , 4 , 8]] 3 1 + x + x − 2 y − 1 1 + y + x (0 , 3) (7 , 1) 6.10 [[48 , 4 , 9]] 3 1 − x + xy 2 1 − y + x 2 (0 , 8) (3 , 3) 6.75 [[52 , 8 , 7]] 3 1 + x + x − 1 y − 2 1 + y + xy − 1 (0 , 13) (2 , 5) 7.54 [[64 , 4 , 11]] 3 1 + x + x − 1 y − 2 1 + y − x − 1 y 2 (0 , 16) (2 , 7) 7.56 [[72 , 4 , 12]] 3 1 − x + y 2 1 − y + x 2 (0 , 12) (3 , 3) 8 [[78 , 6 , 11]] 3 1 + x + x − 1 y 2 1 − y + x − 2 y − 1 (0 , 13) (3 , 4) 9.31 [[104 , 6 , 14]] 3 1 − x + xy − 3 1 − y + x − 2 y 2 (0 , 26) (2 , − 9) 11.31 [[130 , 6 , 16]] 3 1 + x + x 2 y 3 1 + y − x − 2 y 2 (0 , 13) (5 , 3) 11.82 [[144 , 6 , 17]] 3 1 + x + x 2 y 2 1 + y + xy − 3 (0 , 36) (2 , 15) 12.04 [[156 , 10 , 15]] 3 1 + x + x − 1 y 2 1 + y + x − 3 y 2 (0 , 26) (3 , 4) 14.42 [[160 , 12 , 14]] 3 1 − x − x − 2 y 2 1 − y − x − 3 y − 2 (0 , 40) (2 , 7) 14.7 [[192 , 8 , 19]] 3 1 + x + x 2 y − 2 1 + y + x 3 y (0 , 16) (6 , 7) 15.04 [[208 , 8 , 20]] 3 1 − x + xy − 3 1 − y + x − 2 y 2 (0 , 26) (4 , 8) 15.38 [[224 , 6 , 24]] 3 1 + x + x 2 y 2 1 + y + xy − 3 (0 , 56) (2 , 15) 15.43 [[234 , 6 , 25]] 3 1 + x − x 2 y − 6 1 + y + x 2 y 3 (0 , 13) (9 , 2) 16.03 [[240 , 8 , 22]] 3 1 + x + xy − 5 1 + y + x 5 (0 , 60) (2 , 13) 16.13 [[242 , 10 , 22]] 3 1 − x + x 3 y − 3 1 − y + x − 4 (0 , 11) (11 , 4) 20 T ABLE I. Z 3 qutrit LDPC co des with n ≤ 250. The Lau- ren t p olynomials f ( x, y ) and g ( x, y ) sp ecify the translation- in v arian t stabilizers, while the vectors  a 1 and  a 2 define the t wisted b oundary conditions of the torus (see Fig. 1 ). W e list [[ n, k , d ]] 3 suc h that k d 2 /n for each entry in the table exceeds that of all instances with smaller n . W e empha- size that the code distances are obtained using the proba- bilistic algorithm of Ref. [ 91 ]; accordingly , the reported v al- ues of d are upper bounds on the exact distance. F or eac h co de, we sample 5 , 000–10 , 000 information sets and repeat the procedure 1 , 000 times to assess consistency , yielding up- p er b ounds that we b elieve are tigh t in practice. [[ n, k, d ]] 5 f ( x, y ) g ( x, y )  a 1  a 2 kd 2 n [[16 , 4 , 4]] 5 1 + x + 3 y 1 + 3 y + x − 1 y (0 , 4) (2 , 1) 4.00 [[20 , 4 , 5]] 5 1 + 2 x + 4 x − 1 y 1 + 3 y + 4 x (0 , 5) (2 , 1) 5.00 [[24 , 4 , 6]] 5 1 + 2 x + 2 xy 1 + 3 y + x − 1 y (0 , 4) (3 , 2) 6.00 [[30 , 4 , 7]] 5 1 + x + 3 x 2 y − 1 1 + 2 y + 2 x 2 y (0 , 15) (1 , 6) 6.53 [[40 , 4 , 9]] 5 1 + x + 3 x 2 y − 1 1 + 3 y + x 2 y (0 , 4) (5 , 1) 8.10 [[48 , 4 , 10]] 5 1 + x + 4 x − 1 y − 1 1 + y + 2 xy − 1 (0 , 6) (4 , 3) 8.33 [[56 , 4 , 11]] 5 1 + 2 x + 2 x − 2 1 + 2 y + 2 x − 1 y − 1 (0 , 4) (7 , 1) 8.64 [[62 , 6 , 10]] 5 1 + 2 x + 4 y 2 1 + 4 y + 3 x 2 y (0 , 31) (1 , 7) 9.68 [[78 , 10 , 9]] 5 1 + 2 x + 2 x 2 y 2 1 + 3 y + x − 3 (0 , 39) (1 , 5) 10.38 [[80 , 6 , 12]] 5 1 + x + 3 x − 2 1 + 3 y + xy − 2 (0 , 10) (4 , 4) 10.80 [[90 , 4 , 16]] 5 1 + x + x − 2 y 1 + y + x − 1 y 3 (0 , 9) (5 , 1) 11.38 [[96 , 6 , 14]] 5 1 + 2 x + 2 x 2 y 1 + 2 y + 3 x − 1 y 2 (0 , 16) (3 , 6) 12.25 [[114 , 4 , 19]] 5 1 + x + x − 1 y − 2 1 + y + 3 x 3 (0 , 19) (3 , 7) 12.67 [[120 , 6 , 17]] 5 1 + x + 3 y 3 1 + 2 y + 2 xy 3 (0 , 30) (2 , 13) 14.45 [[124 , 6 , 18]] 5 1 + x + 3 x 3 y − 2 1 + 3 y + 4 x 3 y (0 , 31) (2 , 12) 15.68 [[144 , 6 , 20]] 5 1 + x + 3 x − 2 y 1 + y + 3 x 2 y 3 (0 , 9) (8 , 1) 16.67 T ABLE I I. Z 5 qudit LDPC codes with n ≤ 150. Similar to T able I . The deriv ations of Eqs. ( 6 ) and ( 7 ) for Z p qudits fol- lo w directly from the corresp onding pro ofs for the Z 2 case established previously in Refs. [ 90 , 95 ]. The right- hand side of Eq. ( 7 ) can be ev aluated efficien tly using a Gr¨ obner basis computed via Buc hberger’s algorithm [ 96 ]. Searc h for biv ariate bicycle codes o v er Z p . — W e [[ n, k, d ]] 7 f ( x, y ) g ( x, y )  a 1  a 2 kd 2 n [[12 , 4 , 3]] 7 1 + 2 x + 5 y 1 + 2 y + 2 x − 1 (0 , 2) (3 , 0) 3.00 [[14 , 4 , 4]] 7 1 + 2 x + 4 xy 1 + 4 y + 2 xy (0 , 7) (1 , 3) 4.57 [[24 , 4 , 6]] 7 1 + 2 x + 5 y 1 + 2 y + 2 x − 1 (0 , 4) (3 , 2) 6.00 [[28 , 4 , 7]] 7 1 + 5 x + x − 1 y − 1 1 + 5 y + x (0 , 7) (2 , 3) 7.00 [[32 , 4 , 8]] 7 1 + x + 2 x 2 y 1 + 2 y + 3 x − 1 y 2 (0 , 4) (4 , 1) 8.00 [[48 , 4 , 10]] 7 1 + x + 5 x 2 y 1 + 3 y + 3 x − 1 y 2 (0 , 8) (3 , 3) 8.33 [[56 , 4 , 12]] 7 1 + x + 5 xy 1 + 2 y + 4 xy − 2 (0 , 7) (4 , 1) 10.29 [[64 , 4 , 13]] 7 1 + 2 x + 2 x − 1 y − 2 1 + 4 y + 5 xy − 1 (0 , 8) (4 , 1) 10.56 [[70 , 4 , 14]] 7 1 + x + 5 xy − 2 1 + 4 y + 2 x − 2 y − 1 (0 , 7) (5 , 1) 11.20 [[76 , 6 , 12]] 7 1 + x + x − 1 y 2 1 + 2 y + 4 x − 1 y − 1 (0 , 19) (2 , 7) 11.37 [[84 , 4 , 16]] 7 1 + x + x − 1 y − 2 1 + y + 3 x 2 y − 2 (0 , 14) (3 , 5) 12.19 [[96 , 6 , 15]] 7 1 + x + y 2 1 + 4 y + 5 x − 2 (0 , 16) (3 , 5) 14.06 [[98 , 10 , 12]] 7 1 + 2 x + 4 x − 1 y 2 1 + 4 y + 2 x − 2 y − 1 (0 , 7) (7 , 0) 14.69 [[126 , 6 , 18]] 7 1 + x + x 2 y 2 1 + 3 y + 3 x 3 (0 , 21) (3 , 15) 15.43 [[128 , 6 , 19]] 7 1 + x + 5 x 3 y − 2 1 + 2 y + 4 xy − 4 (0 , 16) (4 , 7) 16.92 [[144 , 6 , 21]] 7 1 + x + 3 x 2 y − 2 1 + y + 5 x − 1 y − 2 (0 , 12) (6 , 3) 18.38 T ABLE I I I. Z 7 qudit LDPC co des with n ≤ 150. Similar to T able I . [[ n, k, d ]] 11 f ( x, y ) g ( x, y )  a 1  a 2 kd 2 n [[16 , 4 , 4]] 11 1 + 7 x + 4 xy 1 + 10 y + 3 x − 1 y 2 (0 , 4) (2 , 1) 4.00 [[20 , 4 , 5]] 11 1 + x + 2 xy 1 + 7 y + 10 xy − 1 (0 , 5) (2 , 1) 5.00 [[22 , 4 , 6]] 11 1 + 5 x + 5 x − 1 1 + 8 y + 2 xy − 1 (0 , 11) (1 , 3) 6.55 [[32 , 4 , 8]] 11 1 + 3 x + 10 y 3 1 + 10 y + 3 x − 1 y 2 (0 , 8) (2 , 1) 8.00 [[40 , 4 , 10]] 11 1 + x + 10 x − 1 y − 1 1 + 7 y + 10 xy − 1 (0 , 10) (2 , 4) 10.00 [[48 , 4 , 11]] 11 1 + 3 x + x − 1 y 1 + 6 y + x − 1 y − 1 (0 , 6) (4 , 1) 10.08 [[60 , 4 , 13]] 11 1 + x + 9 x 2 y 2 1 + 5 y + 5 xy − 1 (0 , 5) (6 , 2) 11.27 [[66 , 4 , 14]] 11 1 + 5 x + 5 y 2 1 + 8 y + 2 x 2 (0 , 11) (3 , 6) 11.88 [[72 , 4 , 15]] 11 1 + 2 x + 9 x − 1 y − 2 1 + 5 y + 2 x − 2 y 2 (0 , 12) (3 , 5) 12.50 [[80 , 6 , 14]] 11 1 + x + 2 x − 2 y − 1 1 + 8 y + 7 xy − 2 (0 , 10) (4 , 4) 14.70 [[96 , 6 , 16]] 11 1 + 2 x + 8 xy − 3 1 + 3 y + 7 x − 2 y 2 (0 , 16) (3 , 3) 16.00 [[100 , 6 , 17]] 11 1 + x + x − 2 y − 1 1 + 5 y + 10 x 2 y − 1 (0 , 10) (5 , 5) 17.34 [[120 , 6 , 20]] 11 1 + x + 2 x − 2 y 3 1 + y + 4 x − 1 y − 4 (0 , 10) (6 , 2) 20.00 [[144 , 6 , 23]] 11 1 + 2 x + 8 x − 2 y 2 1 + 2 y + 8 x − 2 y − 3 (0 , 8) (9 , 1) 22.04 T ABLE IV. Z 11 qudit LDPC co des with n ≤ 150. Similar to T able I . in tro duce the gener alize d toric c o de ov er Z p , a particular sub class of biv ariate bicycle (BB) co des, defined b y f ( x, y ) = 1 + r 1 x + r 2 x a y b , g ( x, y ) = 1 + r 3 y + r 4 x c y d , (8) with r i ∈ Z p \ { 0 } and a, b, c, d ∈ Z , as illustrated in Fig. 1(a) . W e systematically searc h for high-performing w eigh t-6 BB codes of the form ( 8 ) o v er Z p qudits with t wisted- p eriodic boundary conditions. F or each even n , we enu- merate all factorizations n = 2 αβ and define a twisted torus by the basis vectors  a 1 = (0 , α ) and  a 2 = ( β , γ ) with 0 ≤ γ < β . W e then enumerate all p olynomials of the form ( 8 ) whose exp onent pairs ( a, b ) and ( c, d ) lie within the fundamen tal parallelogram spanned by  a 1 and  a 2 . F or eac h candidate, we compute the corresponding logical dimension k using Eq. ( 7 ). The computational cost of ev aluating k in Eq. ( 7 ) is essen tially indep en- den t of system size: the b oundary relations y α − 1 and x β y γ − 1 are reduced modulo the ideal ⟨ f , g ⟩ , and only the 4 A B C D E F G H I J K A B C D E F G H I J K 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 (a) [[242 , 10 , 22]] 3 code. A B C D E F A B C D E F 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 (b) [[120 , 6 , 20]] 11 code. FIG. 2. Representativ e stabilizer realizations on t wisted tori. (a) X - and Z -stabilizers of the [[242 , 10 , 22]] 3 co de with f = 1 − x + x 3 y − 3 and g = 1 − y + x − 4 , defined with t wisted b oundary v ectors  a 1 = (0 , 11) and  a 2 = (11 , 4). (b) X - and Z -stabilizers of the [[120 , 6 , 20]] 11 co de with f = 1 + x + 2 x − 2 y 3 and g = 1 + y + 4 x − 1 y − 4 , defined with  a 1 = (0 , 10) and  a 2 = (6 , 2). All other stabilizers are obtained by translations. In eac h panel, v ertices with the same letter (top/bottom) or the same num b er (left/right) are identified, yielding a twisted torus. resulting remainders en ter the Gr¨ obner-basis dimension calculation. F or instances with k > 0, w e then compute the code distance d using the probabilistic algorithm [ 91 ], thereb y obtaining the full [[ n, k , d ]] p parameters. The results for qudit LDPC co des o v er Z p with p = 3 , 5 , 7 , 11 are summarized in T ables I – IV , cov ering p = 3 with n ≤ 250 and p = 5 , 7 , 11 with n ≤ 150. When m ul- tiple co des achiev e the same optimal kd 2 /n , w e select the one with the most local stabilizers as the representativ e example. By comparing the Z 2 results [ 90 ] summarized in Ap- p endix A with our data for Z 3 , Z 5 , Z 7 , and Z 11 , w e obtain the plot in Fig. 3(a) . The vertical axis shows the figure of merit k d 2 /n , while the horizon tal axis shows the minimum num b er of ph ysical qudits n required to ac hieve a given v alue of kd 2 /n . F or each fixed p , the dependence of k d 2 /n on n is ap- pro ximately linear o v er the fitted range, with coefficient of determination R 2 ≈ 0 . 95–0 . 98. Moreo v er, at fixed n we observe that the b est ac hiev able k d 2 /n increases with the qudit dimension p . F or example, at comparable p erformance, the [[120 , 6 , 20]] 11 co de ( k d 2 /n = 20) uses n = 120 qudits, whereas the best Z 2 instance in our com- parison, [[360 , 12 , 24]] 2 ( k d 2 /n = 19 . 2), requires n = 360 qubits. Finally , the linear fits in Fig. 3(a) suggest that the fitted slope of k d 2 /n versus n grows approximately as ln p , as summarized in Fig. 3(b) . Relation to the Bra vyi–P oulin–T erhal T radeoff. — Motiv ated by the trend in k d 2 /n across different primes p in Fig. 3 , w e replot the same data in Fig. 4 as k d 2 /n v ersus (ln p ) n and p erform a linear least-squares fit, ob- taining k d 2 = 0 . 0541 n 2 ln p + 3 . 84 n, (9) for the explored primes p ∈ { 2 , 3 , 5 , 7 , 11 } . At first glance, the quadratic dep endence on n ma y seem to con- tradict the Bra vyi–Poulin–T erhal (BPT) b ound [ 97 , 98 ], whic h states that for geometrically lo cal stabilizer co des in tw o dimensions k d 2 = O ( n ) , (10) where the implicit constan t depends on the in teraction range r and the on-site Hilbert-space dimension. T o make this dependence explicit, w e follo w the BPT argumen t. The lattice is partitioned in to an array of R × R blo c ks suc h that eac h block is correctable, and the blo ck corners are co vered by a region C . By the cleaning lemma, one may c ho ose R = Ω( d/r ). The pro of in Ref. [ 98 ] then implies k ≤ S ( C ) ≤ | C | = O Å nr 2 R 2 ã = O Å nr 4 d 2 ã , (11) where S ( C ) denotes the entanglemen t entrop y of region C . Here, we use the fact that each corner region has linear size O ( r ) (and hence area O ( r 2 )), and that the n umber of blo c ks is n/R 2 . It follows that k d 2 = O ( nr 4 ) . (12) Sup erlinear b eha vior in n is therefore compatible with BPT whenever the in teraction range r increases with system size. In our generalized toric constructions, the geometric diameter of the weigh t-6 stabilizers can gro w with n , dep ending on the chosen exponents in f ( x, y ) and g ( x, y ). This relaxes the BPT constraint and is con- sisten t with the empirical relation ( 9 ). Discussion and future directions. — W e hav e de- v elop ed a top ological-order p erspective on translation- in v arian t Z p stabilizer co des on twisted tori. In this 5 0 50 100 150 200 250 300 350 n 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 k d 2 / n Z 2 q u b i t c o d e s k d 2 / n = 0 . 0 4 0 5 n + 3 . 8 5 , R 2 = 0 . 9 7 6 Z 3 q u t r i t c o d e s k d 2 / n = 0 . 0 5 5 7 n + 4 . 3 0 , R 2 = 0 . 9 4 7 Z 5 q u d i t c o d e s k d 2 / n = 0 . 0 9 0 6 n + 3 . 5 7 , R 2 = 0 . 9 7 6 Z 7 q u d i t c o d e s k d 2 / n = 0 . 1 0 4 7 n + 3 . 5 5 , R 2 = 0 . 9 7 0 Z 1 1 q u d i t c o d e s k d 2 / n = 0 . 1 4 2 4 n + 2 . 8 7 , R 2 = 0 . 9 7 9 (a) k d 2 /n vs. n for the [[ n, k , d ]] p codes in T ables I – V . l n ( 2 ) l n ( 3 ) l n ( 5 ) l n ( 7 ) l n ( 1 1 ) l n ( p ) 0.04 0.06 0.08 0.10 0.12 0.14 Slope y = 0 . 0 5 9 5 4 l n ( p ) - 0 . 0 0 5 4 3 , R 2 = 0 . 9 8 4 9 5 p = 2 p = 3 p = 5 p = 7 p = 1 1 (b) Fitted slop es from panel (a) vs. ln( p ). FIG. 3. Finite-size p erformance across qudit dimensions. (a) [[ n, k , d ]] p LDPC co des from T ables I – V : the metric k d 2 /n is plotted versus block length n for p ∈ { 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 b y the standard errors from the linear fits in panel (a). Linear regression indicates an approximately linear dep endence on ln( p ). 0 50 100 150 200 250 300 350 x = ( l n p ) n 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 y = k d 2 / n p=2 p=3 p=5 p=7 p=11 y = 0 . 0 5 4 1 x + 3 . 8 4 2 9 , R 2 = 0 . 9 6 0 0 4 FIG. 4. Finite-size performance metric k d 2 /n for all code instances listed in T ables I – V , plotted against (ln p ) n for the explored primes p ∈ 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. framew ork, the logical dimension k is determined by the an yon algebra and computed as the dimension of the quotien t ring R /I , where I = ⟨ f ( x, y ) , g ( x, y ) ⟩ is the stabilizer ideal. This ring-theoretic approac h naturally incorp orates t wisted b oundary conditions and, together with Gr¨ obner-basis techniques, enables efficient c harac- terization and systematic searches for new qudit LDPC co des. The high-p erforming instances summarized in T a- bles I – IV pro vide a foundation for further theoretical and hardw are-oriented in vestigations. F uture w ork ma y extend the searc h to larger system sizes, where further impro v ements could emerge. Be- cause the en umeration procedure is fully parallelizable, it can b e scaled to n ≤ 500 or b ey ond using clusters or sup ercomputers, approaching the scale of current exp er- imen tal platforms [ 99 – 104 ]. The primary computational b ottlenec k is distance estimation: for n of a few h undred and d > 25, existing probabilistic distance algorithms t ypically yield only upp er bounds and may become un- reliable. It will also be imp ortan t to v erify whether the empirical relation ( 9 ) persists at larger n and p , and to establish a theoretical understanding of the observ ed n 2 ln p scaling. F urthermore, allo wing higher-w eight stabilizers is an- other natural direction, as suc h constructions ha v e led to impro ved [[ n, k , d ]] parameters in other quan tum LDPC co de families [ 22 , 31 , 105 – 116 ]. Finally , it will be imp ortant to assess the p erformance of these codes under circuit-lev el noise models [ 19 ]. In particular, syndrome-extraction circuits m ust b e opti- mized for low depth, as the circuit-level distance is t ypically smaller than the co de distance. 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[116] Zijian Liang and Y u-An Chen, “Self-dual biv ariate bi- cycle co des with transversal clifford gates,” (2026), arXiv:2510.05211 [quant-ph] . 10 App endix A: Z 2 biv ariate bicycle co des This app endix collects representativ e instances of Z 2 biv ariate bicycle (BB) codes, compiled from Ref. [ 90 ]. W e adopt the polynomial description used there: each co de instance is sp ecified by a pair of Laurent p olynomials f ( x, y ) and g ( x, y ) with Z 2 co efficien ts, together with a c hoice of b oundary iden tifications that render the translation lattice finite. The latter are enco ded by tw o twisted boundary v ectors  a 1 ,  a 2 ∈ Z 2 . F or eac h choice of ( f , g ;  a 1 ,  a 2 ), one obtains a qubit LDPC stabilizer co de with parameters [[ n, k, d ]] 2 , where k is the num ber of logical qubits and d is the code distance of the underlying stabilizer code. Since our goal here is to pro vide concrete test cases spanning a range of performance, we select examples with progressiv ely larger v alues of the figure of merit k d 2 /n . This quan tit y is a conv enien t summary statistic that com bines rate and distance for finite-size comparisons; it do es not b y itself determine p erformance under a sp ecific deco der or circuit-level noise mo del, but it is useful for organizing represen tativ e instances. T able V lists the resulting co de parameters, the defining polynomials f ( x, y ) and g ( x, y ), the boundary vectors  a 1 ,  a 2 , and the corresp onding k d 2 /n v alues. These instances serve as representativ e b enchmarks in the main text for comparing finite-size behavior across differen t co de constructions. [[ n, k , d ]] f ( x, y ) g ( x, y )  a 1  a 2 kd 2 n [[18 , 4 , 4]] 1 + x + xy 1 + y + xy (0 , 3) (3 , 0) 3.56 [[30 , 4 , 6]] 1 + x + x 2 1 + y + x 2 (0 , 3) (5 , 1) 4.8 [[42 , 6 , 6]] 1 + x + xy 1 + y + xy − 1 (0 , 7) (3 , 2) 5.14 [[48 , 4 , 8]] 1 + x + x 2 1 + y + x 2 (0 , 3) (8 , 1) 5.33 [[56 , 6 , 8]] 1 + x + y − 2 1 + y + x − 2 (0 , 7) (4 , 3) 6.86 [[72 , 8 , 8]] 1 + x + x − 1 y 3 1 + y + x 3 y − 1 (0 , 12) (3 , 3) 7.11 [[84 , 6 , 10]] 1 + x + x − 2 1 + y + x − 2 y 2 (0 , 14) (3 , − 6) 7.14 [[90 , 8 , 10]] 1 + x + x − 1 y − 3 1 + y + x 3 y − 1 (0 , 15) (3 , − 6) 8.89 [[120 , 8 , 12]] 1 + x + x − 2 y 1 + y + xy 2 (0 , 10) (6 , 4) 9.6 [[144 , 12 , 12]] 1 + x + x − 1 y − 3 1 + y + x 3 y − 1 (0 , 12) (6 , 0) 12 [[210 , 10 , 16]] 1 + x + x − 3 y 2 1 + y + x − 3 y − 1 (0 , 21) (5 , 10) 12.19 [[248 , 10 , 18]] 1 + x + x − 2 y 1 + y + x − 3 y − 2 (0 , 62) (2 , 25) 13.06 [[254 , 14 , 16]] 1 + x + x − 1 y − 3 1 + y + y − 6 (0 , 127) (1 , 25) 14.11 [[310 , 10 , 22]] 1 + x + x 3 y 2 1 + y + x − 4 y 4 (0 , 31) (5 , 11) 15.61 [[360 , 12 , 24]] 1 + x + x − 1 y 3 1 + y + x 3 y − 1 (0 , 30) (6 , 6) 19.2 T ABLE V. Z 2 qubit LDPC co des compiled from Ref. [ 90 ]. The table rep orts the co de parameters [[ n, k , d ]], the corresp onding p olynomials f ( x, y ) and g ( x, y ), the twisted b oundary vectors  a 1 ,  a 2 , and the metric k d 2 /n .