Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2306.16400 · Physical Review A

Quantum two-block group algebra codes

Hsiang-Ku Lin, Leonid P. Pryadko

Published Feb 5, 2024• 33 citations
arXivHTMLPDF

Abstract

We consider quantum two-block group algebra (2BGA) codes, a family of smallest lifted-product (LP) codes. These codes are related to generalized-bicycle codes, except a cyclic group is replaced with an arbitrary finite group, generally non-Abelian. As special cases, 2BGA codes include a subset of square-matrix LP codes over Abelian groups, including quasicyclic codes, and all square-matrix hypergraph-product codes constructed from a pair of classical group codes. We establish criteria for permutation equivalence of 2BGA codes and give bounds for their parameters, both explicit and in relation to other quantum and classical codes. We also enumerate the optimal parameters of all inequivalent binary connected 2BGA codes with stabilizer generator weights $W\ensuremath{\le}8$, of length $n\ensuremath{\le}100$ for Abelian groups, and $n\ensuremath{\le}200$ for non-Abelian groups.

DOI
10.1103/physreva.109.022407

Related papers

Breaking the bicycle frame: Coset-based quantum LDPC codes

Arda Aydin, Itzhak Tamo, Alexander Barg

Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes constructed using the action of a group on the cosets of its subgroup. This replaces the regular group actions of the earlier two-block constructions and significantly expands the search space, yielding new quantum LDPC codes outside the 2BGA family. Through a computer search, we identify several new quantum LDPC codes, including weight-6 codes with parameters $[[48,8,6]]$, $[[96,8,10]]$, and $[[224,12,16]]$, as well as weight-8 codes with parameters $[[84,16,8]]$, $[[112,16,10]]$, $[[128,16,12]]$, and $[[168,16,15]]$. Furthermore, we introduce a maximally packed syndrome extraction schedule of depth $w+2$, including initialization and measurement steps, for any code with a maximum stabilizer weight of $w$ from our family. Under a standard circuit-level noise model, our codes, when decoded using BP-OSD, perform competitively with BB codes, achieving thresholds of $\approx0.65\%$ for the weight-6 family and $\approx0.35\%$ for the weight-8 family. Finally, we introduce a group-theoretic framework to generate sequences of graph-based covers of 2BGA codes, recovering and extending recent results on code constructions of this type.

quant-pharXiv:2606.17268•Jun 17, 2026•0 citations

Abelian and Non-Abelian Quantum Two-Block Codes

Renyu Wang, Hsiang-Ku Lin, Leonid P. Pryadko

We discuss quantum two-block codes, a large class of CSS codes constructed from two commuting square matrices. Interesting families of such codes are generalized-bicycle (GB) codes and two-block group-algebra (2BGA) codes, where a cyclic group is replaced with an arbitrary finite group, generally non-abelian. We present code construction and give several expressions for code dimension, applicable depending on whether the constituent group is cyclic, abelian, or non-abelian. This gives a simple criterion for an essentially non-abelian 2BGA code guaranteed not to be permutation-equivalent to such a code based on an abelian group. We also give a lower bound on the distance which, in particular, applies to the case when a 2BGA code reduces to a hypergraph-product code constructed from a pair of classical group codes.

arXiv:2305.06890•International Symposium on Turbo Codes and Iterative Information Processing•Sep 4, 2023•6 citations

Bivariate Bicycle Codes over Group Algebras

Chaobin Liu

We study a weighted-shift formulation of bivariate bicycle group-algebra (BBGA) codes over finite group algebras. One binary shift acts by right multiplication and the other by left multiplication; associativity ensures that these shifts commute even for a nonabelian seed group. Bivariate polynomials in the shifts, including mixed monomials, therefore define CSS codes. We give explicit sparsity bounds and distinguish nonabelian seed groups from codes that are intrinsically inequivalent to abelian realizations. For $D_3$, a pair of weighted permutation shifts generates a regular $C_6\times C_{12}$ action and recovers the known $[[144,12,12]]$ and $[[144,14,14]]$ bivariate bicycle codes. We also present a computer-verified $[[144,16,12]]$ code over $A_4\times C_6$, with every check of weight eight. This example belongs to the established nonabelian two-block group-algebra family. Exhaustive low-weight enumeration and an intrinsic stabilizer-support invariant certify that it is inequivalent to any ordinary abelian BB or two-block group-algebra realization, even under qubit permutations combined with local Clifford operations. Finally, a nonuniform weighted-shift construction over $\mathbb{F}_2[C_3]$ yields a $[[18,4,3]]$ code with check weights six and eight. An exact enumeration of its minimum stabilizer supports rules out every ordinary binary 2BGA realization at that length, under the same equivalence. This establishes that the unrestricted weighted-shift formulation extends beyond the ordinary 2BGA class, even with an abelian seed group.

quant-pharXiv:2609.36213•Sep 30, 2026•0 citations

Bicycle Race: small two-block codes on the Pareto frontier

Gideon Lee

Two-block group-algebra codes are among the simplest quantum LDPC codes, and many of them are translation invariant on a twisted torus, which suits neutral-atom arrays and other platforms with movable qubits. We ask how far they go at small sizes when searched systematically across all groups. We use algebraic constraints and screens to design a search pipeline capable of enumerating broad and complete swathes of two-block codes on a laptop. With it we census about $10^8$ two-block codes with up to $320$ qubits at check weights five to nine. Several of these censuses are exhaustive, and support complete statements; for instance, the Gross code is, up to equivalence, the only abelian weight-six two-block code with $n \le 200$ to reach $kd^2/n=12$, no weight-six two-block code with $n \le 300$ beats $[[288,16,16]]$, and the known $[[64,18,8]]$ is the best two-block code by $kd^2/n$ across weights five to nine with $n \le 94$. The censuses are also large enough to give insights into where good codes live - they concentrate in a handful of groups; abelian groups give the best codes at small $n$ and non-abelian groups at large $n$, with the crossover appearing to move out as the check weight grows; and high quality codes become $100$ to $1000$ times rarer per draw from weight six to weights eight and nine. Using the group algebra and the census data to steer a statistical search finds four to fifteen times as many good codes as uniform draws in equal time. Among the codes found are $[[102,20,10]]$ and $[[120,24,10]]$ at check weight nine, which we simulate in noisy syndrome-extraction circuits, and $[[186,20,18]]$, a weight-nine code with every check in a $5\times5$ window. At check weights seven to nine, our best codes exceed the best published $kd^2/n$ with exact distance at the same check weight in most length bands up to $320$ qubits. All codes are available in a public database.

quant-pharXiv:2610.13880•Oct 12, 2026•0 citations

Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search

Liangdong Lu, Guanmin Guo, Yang Liu +1 more

Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes $[[42,12,4]]_2$ and $[[62,12,4]]_2$ and produces a family of codes with competitive figure of merit $kd^2/n$, including $[[66,20,7]]_2$ with $kd^2/n=14.85$, above the bivariate bicycle code $[[144,12,12]]_2$ ($kd^2/n=12$) at less than half the block length, together with $[[46,2,8]]_2$, $[[66,2,9]]_2$, $[[66,4,8]]_2$, $[[66,6,8]]_2$ and, at $n=90$, $[[90,16,6]]_2$, $[[90,18,6]]_2$, $[[90,20,6]]_2$. An exhaustive census at $n=48$ delineates the boundary of this picture: we exhibit a $[[48,10,6]]_2$ code from a minimal $48$-element group (the Aydin--Tamo--Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes $[[48,10,5]]_2$ from the weight-$8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.

cs.ITCross-listedarXiv:2608.09115•Aug 11, 2026•0 citations

Single-shot and two-shot decoding with generalized bicycle codes

Hsiang-Ku Lin, Liu, Xingrui, Pak Kau Lim +1 more

Generalized-bicycle (GB) and more general two-block group-algebra (2BGA) quantum error-correcting codes have naturally redundant minimum-weight stabilizer generators. To use this redundancy, we constructed a large number of ``planar'' 2BGA codes over abelian groups with one and two generators, with each block row of weight 3, relatively large dimensions, distances, and maximum syndrome distance $d_{\rm S}=3$. We simulated the performance of three such codes under phenomenological noise and standard circuit noise, using sliding window sequential decoding protocol covering $T\ge 1$ measurement rounds at a time, based on an in-house binary BP+OSD decoder. While true single-shot decoding ($T=1$) suffers from a significant loss of accuracy, already two-shot ($T=2$) decoding gives nearly the same logical error rates as multi-shot with much larger $T$. Comparison with the same codes but additional stabilizer generators dropped shows that redundancy significantly improves decoding accuracy for all $T\ge 1$.

arXiv:2502.19406•Feb 26, 2025•1 citation
We consider quantum two-block group algebra (2BGA) codes, a family of smallest lifted-product (LP) codes. These codes are related to generalized-bicycle codes, except a cyclic group is replaced with an arbitrary finite group, generally non-Abelian. As special cases, 2BGA codes include a subset of square-matrix LP codes over Abelian groups, including quasicyclic codes, and all square-matrix hypergraph-product codes constructed from a pair of classical group codes. We establish criteria for permutation equivalence of 2BGA codes and give bounds for their parameters, both explicit and in relation to other quantum and classical codes. We also enumerate the optimal parameters of all inequivalent binary connected 2BGA codes with stabilizer generator weights W\ensuremath≤8W\ensuremath{\le}8W\ensuremath≤8, of length n\ensuremath≤100n\ensuremath{\le}100n\ensuremath≤100 for Abelian groups, and n\ensuremath≤200n\ensuremath{\le}200n\ensuremath≤200 for non-Abelian groups.
Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes constructed using the action of a group on the cosets of its subgroup. This replaces the regular group actions of the earlier two-block constructions and significantly expands the search space, yielding new quantum LDPC codes outside the 2BGA family. Through a computer search, we identify several new quantum LDPC codes, including weight-6 codes with parameters [[48,8,6]][[48,8,6]][[48,8,6]], [[96,8,10]][[96,8,10]][[96,8,10]], and [[224,12,16]][[224,12,16]][[224,12,16]], as well as weight-8 codes with parameters [[84,16,8]][[84,16,8]][[84,16,8]], [[112,16,10]][[112,16,10]][[112,16,10]], [[128,16,12]][[128,16,12]][[128,16,12]], and [[168,16,15]][[168,16,15]][[168,16,15]]. Furthermore, we introduce a maximally packed syndrome extraction schedule of depth w+2w+2w+2, including initialization and measurement steps, for any code with a maximum stabilizer weight of www from our family. Under a standard circuit-level noise model, our codes, when decoded using BP-OSD, perform competitively with BB codes, achieving thresholds of ≈0.65%\approx0.65\%≈0.65% for the weight-6 family and ≈0.35%\approx0.35\%≈0.35% for the weight-8 family. Finally, we introduce a group-theoretic framework to generate sequences of graph-based covers of 2BGA codes, recovering and extending recent results on code constructions of this type.
We discuss quantum two-block codes, a large class of CSS codes constructed from two commuting square matrices. Interesting families of such codes are generalized-bicycle (GB) codes and two-block group-algebra (2BGA) codes, where a cyclic group is replaced with an arbitrary finite group, generally non-abelian. We present code construction and give several expressions for code dimension, applicable depending on whether the constituent group is cyclic, abelian, or non-abelian. This gives a simple criterion for an essentially non-abelian 2BGA code guaranteed not to be permutation-equivalent to such a code based on an abelian group. We also give a lower bound on the distance which, in particular, applies to the case when a 2BGA code reduces to a hypergraph-product code constructed from a pair of classical group codes.
We study a weighted-shift formulation of bivariate bicycle group-algebra (BBGA) codes over finite group algebras. One binary shift acts by right multiplication and the other by left multiplication; associativity ensures that these shifts commute even for a nonabelian seed group. Bivariate polynomials in the shifts, including mixed monomials, therefore define CSS codes. We give explicit sparsity bounds and distinguish nonabelian seed groups from codes that are intrinsically inequivalent to abelian realizations. For D3D_3D3​, a pair of weighted permutation shifts generates a regular C6×C12C_6\times C_{12}C6​×C12​ action and recovers the known [[144,12,12]][[144,12,12]][[144,12,12]] and [[144,14,14]][[144,14,14]][[144,14,14]] bivariate bicycle codes. We also present a computer-verified [[144,16,12]][[144,16,12]][[144,16,12]] code over A4×C6A_4\times C_6A4​×C6​, with every check of weight eight. This example belongs to the established nonabelian two-block group-algebra family. Exhaustive low-weight enumeration and an intrinsic stabilizer-support invariant certify that it is inequivalent to any ordinary abelian BB or two-block group-algebra realization, even under qubit permutations combined with local Clifford operations. Finally, a nonuniform weighted-shift construction over F2[C3]\mathbb{F}_2[C_3]F2​[C3​] yields a [[18,4,3]][[18,4,3]][[18,4,3]] code with check weights six and eight. An exact enumeration of its minimum stabilizer supports rules out every ordinary binary 2BGA realization at that length, under the same equivalence. This establishes that the unrestricted weighted-shift formulation extends beyond the ordinary 2BGA class, even with an abelian seed group.
Two-block group-algebra codes are among the simplest quantum LDPC codes, and many of them are translation invariant on a twisted torus, which suits neutral-atom arrays and other platforms with movable qubits. We ask how far they go at small sizes when searched systematically across all groups. We use algebraic constraints and screens to design a search pipeline capable of enumerating broad and complete swathes of two-block codes on a laptop. With it we census about 10810^8108 two-block codes with up to 320320320 qubits at check weights five to nine. Several of these censuses are exhaustive, and support complete statements; for instance, the Gross code is, up to equivalence, the only abelian weight-six two-block code with n≤200n \le 200n≤200 to reach kd2/n=12kd^2/n=12kd2/n=12, no weight-six two-block code with n≤300n \le 300n≤300 beats [[288,16,16]][[288,16,16]][[288,16,16]], and the known [[64,18,8]][[64,18,8]][[64,18,8]] is the best two-block code by kd2/nkd^2/nkd2/n across weights five to nine with n≤94n \le 94n≤94. The censuses are also large enough to give insights into where good codes live - they concentrate in a handful of groups; abelian groups give the best codes at small nnn and non-abelian groups at large nnn, with the crossover appearing to move out as the check weight grows; and high quality codes become 100100100 to 100010001000 times rarer per draw from weight six to weights eight and nine. Using the group algebra and the census data to steer a statistical search finds four to fifteen times as many good codes as uniform draws in equal time. Among the codes found are [[102,20,10]][[102,20,10]][[102,20,10]] and [[120,24,10]][[120,24,10]][[120,24,10]] at check weight nine, which we simulate in noisy syndrome-extraction circuits, and [[186,20,18]][[186,20,18]][[186,20,18]], a weight-nine code with every check in a 5×55\times55×5 window. At check weights seven to nine, our best codes exceed the best published kd2/nkd^2/nkd2/n with exact distance at the same check weight in most length bands up to 320320320 qubits. All codes are available in a public database.
Quantum Bicycle LDPC Codes with High kd2/nkd^2/nkd2/n from Divisor-Driven Search
Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes [[42,12,4]]2[[42,12,4]]_2[[42,12,4]]2​ and [[62,12,4]]2[[62,12,4]]_2[[62,12,4]]2​ and produces a family of codes with competitive figure of merit kd2/nkd^2/nkd2/n, including [[66,20,7]]2[[66,20,7]]_2[[66,20,7]]2​ with kd2/n=14.85kd^2/n=14.85kd2/n=14.85, above the bivariate bicycle code [[144,12,12]]2[[144,12,12]]_2[[144,12,12]]2​ (kd2/n=12kd^2/n=12kd2/n=12) at less than half the block length, together with [[46,2,8]]2[[46,2,8]]_2[[46,2,8]]2​, [[66,2,9]]2[[66,2,9]]_2[[66,2,9]]2​, [[66,4,8]]2[[66,4,8]]_2[[66,4,8]]2​, [[66,6,8]]2[[66,6,8]]_2[[66,6,8]]2​ and, at n=90n=90n=90, [[90,16,6]]2[[90,16,6]]_2[[90,16,6]]2​, [[90,18,6]]2[[90,18,6]]_2[[90,18,6]]2​, [[90,20,6]]2[[90,20,6]]_2[[90,20,6]]2​. An exhaustive census at n=48n=48n=48 delineates the boundary of this picture: we exhibit a [[48,10,6]]2[[48,10,6]]_2[[48,10,6]]2​ code from a minimal 484848-element group (the Aydin--Tamo--Barg realization uses 727272 elements), and prove that distance 555 forces a stabilizer-rank loss, which excludes [[48,10,5]]2[[48,10,5]]_2[[48,10,5]]2​ from the weight-888 symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.
Generalized-bicycle (GB) and more general two-block group-algebra (2BGA) quantum error-correcting codes have naturally redundant minimum-weight stabilizer generators. To use this redundancy, we constructed a large number of ``planar'' 2BGA codes over abelian groups with one and two generators, with each block row of weight 3, relatively large dimensions, distances, and maximum syndrome distance dS=3d_{\rm S}=3dS​=3. We simulated the performance of three such codes under phenomenological noise and standard circuit noise, using sliding window sequential decoding protocol covering T≥1T\ge 1T≥1 measurement rounds at a time, based on an in-house binary BP+OSD decoder. While true single-shot decoding (T=1T=1T=1) suffers from a significant loss of accuracy, already two-shot (T=2T=2T=2) decoding gives nearly the same logical error rates as multi-shot with much larger TTT. Comparison with the same codes but additional stabilizer generators dropped shows that redundancy significantly improves decoding accuracy for all T≥1T\ge 1T≥1.