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arXiv:2608.12509

Quantum Tanner Codes at Moderate Blocklength

Feroz Ahmed Mian, Vaishnavi L. Addala, Arman Meraj, Adhiraj Chadha, Stefan Krastanov

quant-phSubmitted Aug 12, 2026• Announced Aug 14, 2026• 0 citations
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Abstract

We present explicit constructions of quantum Tanner (QT) codes with good rate and distance, obtained through two complementary approaches: the left-right Cayley complex (LRCC) description and the "lifting" perspective, in which a seed Calderbank-Shor-Steane (CSS) code is lifted by commuting left-right regular actions of a finite group $\mathcal{G}$. Through an extensive search over non-abelian groups from GAP's SmallGrp library, we investigate the moderate-blocklength regime ($n \in [500,1000]$) and identify several new code instances with distance upper bounds exceeding $20$. These include $[[480,8,(\leq 21,\leq 21)]]$, $[[504,4,(\leq 36,\leq 27)]]$, $[[672,4,(\leq 48,\leq 28)]]$, $[[720,6,(\leq 30,\leq 30)]]$, and $[[864,8,(\leq 39,\leq 31)]]$, with these bounds obtained using up to $350$ million trials of sQetch, a randomized distance estimator. The code instances presented have check weights ranging from $9$ to $20$. Using the Tesseract decoder, we estimate pseudo-thresholds of $3.6\%$-$4.6\%$ under phenomenological noise and $0.14\%$-$0.27\%$ under circuit-level noise, comparable to prior results at shorter code lengths. We also provide QuantumExpanders$.$jl, an open-source Julia library for constructing QT codes and explicit constructions of Ramanujan graphs.

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Small quantum Tanner codes from left--right Cayley complexes

Anthony Leverrier, Wouter Rozendaal, Gilles Zémor

Quantum Tanner codes are a class of quantum low-density parity-check codes that provably display a linear minimum distance and a constant encoding rate in the asymptotic limit. When built from left--right Cayley complexes, they can be described through a lifting procedure and a base code, which we characterize. We also compute the dimension of quantum Tanner codes when the right degree of the complex is 2. Finally, we perform an extensive search over small groups and identify instances of quantum Tanner codes with parameters $[[144,12,11]]$, $[[432,20,\leq 22]]$ and $[[576,28,\leq 24]]$ for generators of weight 9.

quant-pharXiv:2512.20532•Dec 24, 2025•0 citations

High-Girth Regular Quantum LDPC Codes from Affine-Coset Structures

Koki Okada, Kenta Kasai

We construct a quantum low-density parity-check code family from a length-$512$ Calderbank--Shor--Steane base matrix pair. The base pair is permutation-equivalent to the known SPC(3) product CSS code, and the present affine-coset description gives a direct proof that both Tanner graphs are $(3,8)$-regular with girth $8$. The base code has parameters $[[512,174,8]]$. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor $P=32$, which has parameters $[[16384,4142,18\le d\le32]]$, under the code-capacity depolarizing model. Complete enumeration excludes nontrivial logical representatives of weight at most $16$ on both sides, giving $d_X,d_Z\ge18$. Explicit weight-$32$ non-stabilizer logical representatives give the upper bounds $d_X,d_Z\le32$. A belief-propagation decoder with post-processing achieved frame error rate about $10^{-8}$ at $p=0.085$; an independently observed logical residual of weight $40$ is consistent with the sharper structural bound.

quant-pharXiv:2604.20838•Apr 23, 2026•0 citations

Moderate-length lifted quantum Tanner codes

Virgile Guémard, Gilles Zémor

We introduce new families of quantum Tanner codes, a class of quantum codes that first appeared in the work of Leverrier and Zémor (FOCS 2022). These codes are built from two classical Tanner codes, for which the underlying graphs are extracted from coverings of 2D geometrical complexes, and the local linear codes are tensor-products of cyclic or double-circulant linear codes. The advantage of code lifting is that, for any lift of odd index $t$ of an $[[n,k,d]]$-code, we can adapt the study of the transfer homomorphism arising in cellular homology to describe symmetries of its logical operators and to establish that its dimension is lower bounded by $k$, and its distance is upper bounded by $t\cdot d$. Moreover, when the dimension of the lifted code is equal to $k$, its distance is lower bounded by $d$. These parameter bounds also apply to the previous methods of code lifting of Guémard (IEEE Trans. Inf. Theory, 2025). Finally, We present several explicit families, and identify instances of moderate length quantum codes which are degenerate, have low check weight, and whose distance surpasses the square root of the code length. Among them, we report the existence of a $[[96,2,12]]$-code whose distance growth saturates our bound, and for which half of the checks are of weight 8 and the other half of weight 4.

arXiv:2502.20297•Feb 27, 2025•0 citations

High-Girth Regular Quantum LDPC Codes from Square-Base Hypergraph Products via CPM Lifts

Koki Okada, Kenta Kasai

We study square-base Calderbank--Shor--Steane (CSS) hypergraph-product codes as a finite-length class for regular high-girth quantum low-density parity-check (LDPC) design. For base matrices of small column weight, we give checkable conditions for regularity, rank deficiency, and short-cycle exclusion, and we present explicit column-weight-three and column-weight-four examples with Tanner girth 6 and 8. We also analyze circulant permutation matrix (CPM) lifts of this class. Using the standard voltage-sum criterion, we identify orthogonality-forced Tanner 8-cycles and show that CPM lifting cannot raise the Tanner girth beyond 8 when these cycles are present. As a representative finite-length instance, a randomized CPM lift of the girth-8 base construction gives a $[[28800,62,18\le d\le192]]$ girth-8 $(3,6)$-regular CSS-LDPC code. Complete enumeration excludes nontrivial logical supports of weight at most $16$ on both sides, giving $d_X,d_Z\ge18$; explicit weight-$192$ representatives give $d_X,d_Z\le192$. Under degeneracy-aware belief-propagation decoding with optional ordered-statistics-decoding-lite post-processing, this code produced zero decoding failures in $2.993\times 10^8$ independent trials at depolarizing probability $p=0.1402$; the Wilson 95\% upper confidence bound is $1.28\times 10^{-8}$.

quant-pharXiv:2604.27817•May 1, 2026•0 citations

Lifts of Quantum CSS Codes - New Constructions Beyond Product Codes

Virgile Guémard

In classical coding theory, the lift of a code is a practical and natural method for generatingfamilies of LDPC codes from a single chosen input code. This strategy has proven to bevery successful in producing LDPC codes of various blocklengths, reaching good parametersand approaching the capacity of certain channels in the asymptotic regime. It is also nowapplied in the 5G New Radio [RK18].It is widely believed that a quantum computer will integrate a quantum error correctingscheme, and the last three decades have seen the emergence of a variety of code-basedquantum error correction known as CSS codes. It is natural to ask whether we can performlifts of quantum codes, with the hope of having the same benefits as for classical codes.However, the method to lift linear codes cannot be directly applied to quantum CSS codes,as the constituent linear codes usually lose their orthogonality. This dissertation presentsthree new results on lifts of quantum CSS codes.Firstly, we propose a general notion of lift for quantum CSS codes. It is based on theexistence of a canonical complex associated to any CSS code, that we call the Tanner cone-complex, and over which we generate covering spaces. We show that our notion generalizesclassical code lifting and generate important CSS code families. Our main application ofthe lift is to devise new types of quantum CSS codes. So far, most seriously consideredconstructions rely on topology or some kind of product operations. Here, we intend toexplore beyond these paradigmatic ideas and for that we construct new non-topologicaland non-product CSS codes. We then apply the prescription to generate their lifts andshow that this procedure can improve their relative parameters, reaching parameter regionsnot explored prior to this work.Secondly, we introduce new families of quantum Tanner codes, a class of quantum codeswhich first appeared in the work of Leverrier and Zémor [LZ22]. We build these codesby generalizing the lift of classical Tanner codes, based on graph coverings, to the case ofCSS codes, based on coverings of 2D geometrical complexes. We present several explicitfamilies, and identify instances of moderate length quantum codes which are degenerate,have low check weight, and for which the distance is greater than the square root of thecode length.Lastly, we present a topological approach to lifting a quantum CSS code. Our originaldefinition of CSS code lifting was inspired by the work of Freedman and Hastings [FH21],which associates CSS codes with handlebodies. This thesis is a good opportunity to showhow the handlebody realization of a code can also be used to perform code lifting. Weprovide a detailed discussion of why in many cases this is essentially equivalent to ourprevious approach.

•Nov 28, 2025•0 citations

Coset Lifted Product Codes: Constructing Quantum LDPC Codes from Non-Normal Subgroup Actions

Rajendra Prasad Rajpurohit, V. Lalitha, Shayan Srinivasa Garani

We introduce coset lifted product (CLP) codes, a family of quantum LDPC codes that combine the protograph structure of lifted product codes with the coset action of Aydin, Tamo, and Barg \cite{AydinTamoBarg2026}. Given a finite group $G$, a subgroup $H \le G$ that need not be normal, a protograph $A$ over $\mathbb F_2[G]$, and a protograph $B$ over $\mathbb F_2[N_G(H)]$, the two protographs act on the cosets $G/H$ from the left and from the right to produce quantum CSS codes. Further, for groups of odd order, we give an explicit formula for the dimension as a sum over the irreducible representations $ρ$ of $G$ of hypergraph product dimensions. A search over $14,843$ pairs $(G,H)$ with $H$ restricted to be non-normal and $120 \le n \le 400$ yields certified codes that exceed the best published value of $Q=kd^2/n$ at check weights $6,7$ and $9$, i.e., we obtain $[[360,20,20]]$ at $w=6$, a $[[360,24,20]]$ code at $w=7$, and codes $[[252,16,18]]$ and $[[288,16,20]]$ at $w=9$. Our search also produce high-distance codes with parameters $[[168,10,14]], [[240,8,20]]$ at $w=6$ and $[[216,8,18]]$ at $w=7$.

quant-pharXiv:2610.12782•Oct 12, 2026•0 citations
We present explicit constructions of quantum Tanner (QT) codes with good rate and distance, obtained through two complementary approaches: the left-right Cayley complex (LRCC) description and the "lifting" perspective, in which a seed Calderbank-Shor-Steane (CSS) code is lifted by commuting left-right regular actions of a finite group G\mathcal{G}G. Through an extensive search over non-abelian groups from GAP's SmallGrp library, we investigate the moderate-blocklength regime (n∈[500,1000]n \in [500,1000]n∈[500,1000]) and identify several new code instances with distance upper bounds exceeding 202020. These include [[480,8,(≤21,≤21)]][[480,8,(\leq 21,\leq 21)]][[480,8,(≤21,≤21)]], [[504,4,(≤36,≤27)]][[504,4,(\leq 36,\leq 27)]][[504,4,(≤36,≤27)]], [[672,4,(≤48,≤28)]][[672,4,(\leq 48,\leq 28)]][[672,4,(≤48,≤28)]], [[720,6,(≤30,≤30)]][[720,6,(\leq 30,\leq 30)]][[720,6,(≤30,≤30)]], and [[864,8,(≤39,≤31)]][[864,8,(\leq 39,\leq 31)]][[864,8,(≤39,≤31)]], with these bounds obtained using up to 350350350 million trials of sQetch, a randomized distance estimator. The code instances presented have check weights ranging from 999 to 202020. Using the Tesseract decoder, we estimate pseudo-thresholds of 3.6%3.6\%3.6%-4.6%4.6\%4.6% under phenomenological noise and 0.14%0.14\%0.14%-0.27%0.27\%0.27% under circuit-level noise, comparable to prior results at shorter code lengths. We also provide QuantumExpanders...jl, an open-source Julia library for constructing QT codes and explicit constructions of Ramanujan graphs.
Quantum Tanner codes are a class of quantum low-density parity-check codes that provably display a linear minimum distance and a constant encoding rate in the asymptotic limit. When built from left--right Cayley complexes, they can be described through a lifting procedure and a base code, which we characterize. We also compute the dimension of quantum Tanner codes when the right degree of the complex is 2. Finally, we perform an extensive search over small groups and identify instances of quantum Tanner codes with parameters [[144,12,11]][[144,12,11]][[144,12,11]], [[432,20,≤22]][[432,20,\leq 22]][[432,20,≤22]] and [[576,28,≤24]][[576,28,\leq 24]][[576,28,≤24]] for generators of weight 9.
We construct a quantum low-density parity-check code family from a length-512512512 Calderbank--Shor--Steane base matrix pair. The base pair is permutation-equivalent to the known SPC(3) product CSS code, and the present affine-coset description gives a direct proof that both Tanner graphs are (3,8)(3,8)(3,8)-regular with girth 888. The base code has parameters [[512,174,8]][[512,174,8]][[512,174,8]]. We then apply circulant permutation matrix (CPM) lifts. The main decoding experiment uses the CPM-lifted code with lift factor P=32P=32P=32, which has parameters [[16384,4142,18≤d≤32]][[16384,4142,18\le d\le32]][[16384,4142,18≤d≤32]], under the code-capacity depolarizing model. Complete enumeration excludes nontrivial logical representatives of weight at most 161616 on both sides, giving dX,dZ≥18d_X,d_Z\ge18dX​,dZ​≥18. Explicit weight-323232 non-stabilizer logical representatives give the upper bounds dX,dZ≤32d_X,d_Z\le32dX​,dZ​≤32. A belief-propagation decoder with post-processing achieved frame error rate about 10−810^{-8}10−8 at p=0.085p=0.085p=0.085; an independently observed logical residual of weight 404040 is consistent with the sharper structural bound.
We introduce new families of quantum Tanner codes, a class of quantum codes that first appeared in the work of Leverrier and Zémor (FOCS 2022). These codes are built from two classical Tanner codes, for which the underlying graphs are extracted from coverings of 2D geometrical complexes, and the local linear codes are tensor-products of cyclic or double-circulant linear codes. The advantage of code lifting is that, for any lift of odd index ttt of an [[n,k,d]][[n,k,d]][[n,k,d]]-code, we can adapt the study of the transfer homomorphism arising in cellular homology to describe symmetries of its logical operators and to establish that its dimension is lower bounded by kkk, and its distance is upper bounded by t⋅dt\cdot dt⋅d. Moreover, when the dimension of the lifted code is equal to kkk, its distance is lower bounded by ddd. These parameter bounds also apply to the previous methods of code lifting of Guémard (IEEE Trans. Inf. Theory, 2025). Finally, We present several explicit families, and identify instances of moderate length quantum codes which are degenerate, have low check weight, and whose distance surpasses the square root of the code length. Among them, we report the existence of a [[96,2,12]][[96,2,12]][[96,2,12]]-code whose distance growth saturates our bound, and for which half of the checks are of weight 8 and the other half of weight 4.
We study square-base Calderbank--Shor--Steane (CSS) hypergraph-product codes as a finite-length class for regular high-girth quantum low-density parity-check (LDPC) design. For base matrices of small column weight, we give checkable conditions for regularity, rank deficiency, and short-cycle exclusion, and we present explicit column-weight-three and column-weight-four examples with Tanner girth 6 and 8. We also analyze circulant permutation matrix (CPM) lifts of this class. Using the standard voltage-sum criterion, we identify orthogonality-forced Tanner 8-cycles and show that CPM lifting cannot raise the Tanner girth beyond 8 when these cycles are present. As a representative finite-length instance, a randomized CPM lift of the girth-8 base construction gives a [[28800,62,18≤d≤192]][[28800,62,18\le d\le192]][[28800,62,18≤d≤192]] girth-8 (3,6)(3,6)(3,6)-regular CSS-LDPC code. Complete enumeration excludes nontrivial logical supports of weight at most 161616 on both sides, giving dX,dZ≥18d_X,d_Z\ge18dX​,dZ​≥18; explicit weight-192192192 representatives give dX,dZ≤192d_X,d_Z\le192dX​,dZ​≤192. Under degeneracy-aware belief-propagation decoding with optional ordered-statistics-decoding-lite post-processing, this code produced zero decoding failures in 2.993×1082.993\times 10^82.993×108 independent trials at depolarizing probability p=0.1402p=0.1402p=0.1402; the Wilson 95\% upper confidence bound is 1.28×10−81.28\times 10^{-8}1.28×10−8.
In classical coding theory, the lift of a code is a practical and natural method for generatingfamilies of LDPC codes from a single chosen input code. This strategy has proven to bevery successful in producing LDPC codes of various blocklengths, reaching good parametersand approaching the capacity of certain channels in the asymptotic regime. It is also nowapplied in the 5G New Radio [RK18].It is widely believed that a quantum computer will integrate a quantum error correctingscheme, and the last three decades have seen the emergence of a variety of code-basedquantum error correction known as CSS codes. It is natural to ask whether we can performlifts of quantum codes, with the hope of having the same benefits as for classical codes.However, the method to lift linear codes cannot be directly applied to quantum CSS codes,as the constituent linear codes usually lose their orthogonality. This dissertation presentsthree new results on lifts of quantum CSS codes.Firstly, we propose a general notion of lift for quantum CSS codes. It is based on theexistence of a canonical complex associated to any CSS code, that we call the Tanner cone-complex, and over which we generate covering spaces. We show that our notion generalizesclassical code lifting and generate important CSS code families. Our main application ofthe lift is to devise new types of quantum CSS codes. So far, most seriously consideredconstructions rely on topology or some kind of product operations. Here, we intend toexplore beyond these paradigmatic ideas and for that we construct new non-topologicaland non-product CSS codes. We then apply the prescription to generate their lifts andshow that this procedure can improve their relative parameters, reaching parameter regionsnot explored prior to this work.Secondly, we introduce new families of quantum Tanner codes, a class of quantum codeswhich first appeared in the work of Leverrier and Zémor [LZ22]. We build these codesby generalizing the lift of classical Tanner codes, based on graph coverings, to the case ofCSS codes, based on coverings of 2D geometrical complexes. We present several explicitfamilies, and identify instances of moderate length quantum codes which are degenerate,have low check weight, and for which the distance is greater than the square root of thecode length.Lastly, we present a topological approach to lifting a quantum CSS code. Our originaldefinition of CSS code lifting was inspired by the work of Freedman and Hastings [FH21],which associates CSS codes with handlebodies. This thesis is a good opportunity to showhow the handlebody realization of a code can also be used to perform code lifting. Weprovide a detailed discussion of why in many cases this is essentially equivalent to ourprevious approach.
We introduce coset lifted product (CLP) codes, a family of quantum LDPC codes that combine the protograph structure of lifted product codes with the coset action of Aydin, Tamo, and Barg \cite{AydinTamoBarg2026}. Given a finite group GGG, a subgroup H≤GH \le GH≤G that need not be normal, a protograph AAA over F2[G]\mathbb F_2[G]F2​[G], and a protograph BBB over F2[NG(H)]\mathbb F_2[N_G(H)]F2​[NG​(H)], the two protographs act on the cosets G/HG/HG/H from the left and from the right to produce quantum CSS codes. Further, for groups of odd order, we give an explicit formula for the dimension as a sum over the irreducible representations ρρρ of GGG of hypergraph product dimensions. A search over 14,84314,84314,843 pairs (G,H)(G,H)(G,H) with HHH restricted to be non-normal and 120≤n≤400120 \le n \le 400120≤n≤400 yields certified codes that exceed the best published value of Q=kd2/nQ=kd^2/nQ=kd2/n at check weights 6,76,76,7 and 999, i.e., we obtain [[360,20,20]][[360,20,20]][[360,20,20]] at w=6w=6w=6, a [[360,24,20]][[360,24,20]][[360,24,20]] code at w=7w=7w=7, and codes [[252,16,18]][[252,16,18]][[252,16,18]] and [[288,16,20]][[288,16,20]][[288,16,20]] at w=9w=9w=9. Our search also produce high-distance codes with parameters [[168,10,14]],[[240,8,20]][[168,10,14]], [[240,8,20]][[168,10,14]],[[240,8,20]] at w=6w=6w=6 and [[216,8,18]][[216,8,18]][[216,8,18]] at w=7w=7w=7.