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:2606.17268

Breaking the bicycle frame: Coset-based quantum LDPC codes

Arda Aydin, Itzhak Tamo, Alexander Barg

quant-phcs.ITSubmitted Jun 15, 2026• Announced Jun 17, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

Related papers

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

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

Mirror codes: High-threshold quantum LDPC codes beyond the CSS regime

Andrey Boris Khesin, Jonathan Z. Lu

The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group $G$ and two subsets of $G$ whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find $[[ 60, 4, 10 ]], [[ 36, 6, 6 ]], [[ 48, 8, 6 ]]$, and $[[ 85, 8, 9 ]]$ codes, all of which are not CSS; we also find several weight-7 codes with $kd > n$. Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of $0.2\%$, approximately matching that of the $[[ 144, 12, 12 ]]$ BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.

quant-pharXiv:2603.05496•Mar 6, 2026•0 citations

Quantum two-block group algebra codes

Hsiang-Ku Lin, Leonid P. Pryadko

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.

arXiv:2306.16400•Physical Review A•Feb 5, 2024•33 citations

Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark

Liangdong Lu, Ruipan Yang, Guanmin Guo

Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-$6$ checks, exemplified by $[[144,12,12]]$ with $kd^2/n=12$. We develop the algebraic structure theory of BB-type codes with weight-$8$ checks (weight-$4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula $k=2\dim R/(A,B)$ (forcing even $k$), a $4\ell m$-element symmetry group on generator pairs, an $X/Z$ distance equality $d_X=d_Z$, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At $n=144$ the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: $[[144,6,d\ge 15]]$ exceeds the benchmark distance $12$ (certified $d\ge 15$), $[[144,10,12]]$ reaches it with weight-$8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at $kd^2/n=11.11$ ($7.4\%$ below benchmark) while decoding no worse. At $n=72$, $[[72,14,8]]$ attains $kd^2/n=12.44$---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-$8$ codes at $\approx 0.1\%$ pseudo-threshold versus $\approx 0.4\%$ for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.

quant-pharXiv:2609.06572•Sep 8, 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
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.
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.
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.
The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group GGG and two subsets of GGG whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find [[60,4,10]],[[36,6,6]],[[48,8,6]][[ 60, 4, 10 ]], [[ 36, 6, 6 ]], [[ 48, 8, 6 ]][[60,4,10]],[[36,6,6]],[[48,8,6]], and [[85,8,9]][[ 85, 8, 9 ]][[85,8,9]] codes, all of which are not CSS; we also find several weight-7 codes with kd>nkd > nkd>n. Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of 0.2%0.2\%0.2%, approximately matching that of the [[144,12,12]][[ 144, 12, 12 ]][[144,12,12]] BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.
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.
Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-666 checks, exemplified by [[144,12,12]][[144,12,12]][[144,12,12]] with kd2/n=12kd^2/n=12kd2/n=12. We develop the algebraic structure theory of BB-type codes with weight-888 checks (weight-444 generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula k=2dim⁡R/(A,B)k=2\dim R/(A,B)k=2dimR/(A,B) (forcing even kkk), a 4ℓm4\ell m4ℓm-element symmetry group on generator pairs, an X/ZX/ZX/Z distance equality dX=dZd_X=d_ZdX​=dZ​, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At n=144n=144n=144 the pipeline returns a census of 535353 codes whose strongest members surpass the BB benchmark: [[144,6,d≥15]][[144,6,d\ge 15]][[144,6,d≥15]] exceeds the benchmark distance 121212 (certified d≥15d\ge 15d≥15), [[144,10,12]][[144,10,12]][[144,10,12]] reaches it with weight-888 checks, and [[144,16,10]][[144,16,10]][[144,16,10]] encodes a third more logical qubits at kd2/n=11.11kd^2/n=11.11kd2/n=11.11 (7.4%7.4\%7.4% below benchmark) while decoding no worse. At n=72n=72n=72, [[72,14,8]][[72,14,8]][[72,14,8]] attains kd2/n=12.44kd^2/n=12.44kd2/n=12.44---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-888 codes at ≈0.1%\approx 0.1\%≈0.1% pseudo-threshold versus ≈0.4%\approx 0.4\%≈0.4% for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.
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.