arXiv:2605.14173

Univariate Bicycle Quantum LDPC Codes: Explicit Logical Structure and Distance Bounds

Sheida Rabeti, Hessam Mahdavifar

Published May 13, 2026• 0 citations

Abstract

We introduce univariate bicycle (UB) codes, a structured subclass of generalized bicycle (GB) quantum low-density parity-check (LDPC) codes obtained via a Frobenius relation. This construction reduces the code design space from a two-polynomial search in GB codes to a single-polynomial search, while preserving sparsity. We provide an explicit algebraic characterization of the logical coset spaces by constructing a basis for the logical quotient space, yielding a complete parametrization of logical operators. Leveraging this structure, we derive upper bounds on the minimum distance by relating structured logical representatives to cycle-density properties of associated circulant matrices. Finally, simulation results for short- to medium-length UB codes (block lengths ranging from a few hundred to approximately $10^3$) demonstrate competitive performance relative to existing GB and bivariate bicycle (BB) codes despite the additional algebraic restriction.

DOI: 10.48550/arxiv.2605.14173

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arXiv:2605.14173

Univariate Bicycle Quantum LDPC Codes: Explicit Logical Structure and Distance Bounds

Sheida Rabeti, Hessam Mahdavifar

Published May 13, 2026• 0 citations

arXiv HTML PDF

Abstract

10^3

) demonstrate competitive performance relative to existing GB and bivariate bicycle (BB) codes despite the additional algebraic restriction.

DOI: 10.48550/arxiv.2605.14173

Related papers

Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes

Mohammad Rowshan, Simon Devitt

Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the $Z$ -logical quotient $K/S$ over $\mathbb F_2[x,y]/(x^\ell-1,y^m-1)$ and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity $d_Z=\min(d_{\mathcal A},d_{\mathcal C})$ . Using the Frobenius structure of the finite group algebra, we prove $r_{\mathcal A}=r_{\mathcal C}=k/2$ for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances $4,6,10,10,12,18$ of the six standard BB codes of lengths $18$ to $288$ and enumerates every minimum-weight logical operator. The resulting census shows that $[[108,8,10]]$ is the only one of the six whose distance is attained in a single component, with $d_{\mathcal C}=10$ and $d_{\mathcal A}=12$ .Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the $Z$-logical quotient $K/S$ over $\mathbb F_2[x,y]/(x^\ell-1,y^m-1)$ and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity $d_Z=\min(d_{\mathcal A},d_{\mathcal C})$. Using the Frobenius structure of the finite group algebra, we prove $r_{\mathcal A}=r_{\mathcal C}=k/2$ for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances $4,6,10,10,12,18$ of the six standard BB codes of lengths $18$ to $288$ and enumerates every minimum-weight logical operator. The resulting census shows that $[[108,8,10]]$ is the only one of the six whose distance is attained in a single component, with $d_{\mathcal C}=10$ and $d_{\mathcal A}=12$.

quant-pharXiv:2609.39395Oct 1, 20260 citations

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

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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.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:2608.09115Aug 11, 20260 citations

List Decoding and New Bicycle Code Constructions for Quantum LDPC Codes

Rabeti, Sheida, Mahdavifar, Hessam

In this paper, we propose a new decoder, called the Multiple-Bases Belief-Propagation List Decoder (MBBP-LD), for Quantum Low-Density Parity-Check (QLDPC) codes. It extends the Multiple-Bases Belief-Propagation (MBBP) framework, originally developed for classical cyclic LDPC codes. The proposed method preserves the linear-time complexity of standard BP decoder while improving the logical error rate. To further reduce the logical error rate, a new decision rule is introduced for the post-processing list decoder, outperforming the conventional least-metric selector (LMS) criterion. For the recently developed and implemented bivariate bicycle (BB) code with parameters $[[144,12,12]]$ , our proposed MBBP-LD decoder achieves up to 40\% lower logical error rate compared to the state-of-the-art decoder for short QLDPC codes, i.e., BP with ordered-statistics decoding (BP-OSD), while retaining the linear-time complexity of the plain BP decoder. In addition, we explore a new subclass of BB codes, that we refer to as the univariate bicycle (UB) codes, specifically with lower-weight parity checks ( $w=6,8$ ). This reduces the polynomial search space for the code compared to general BB codes, i.e., by reducing the search space over two polynomial components in BB codes to just a single polynomial component in UB codes. Simulations demonstrate the promising performance of these codes under various types of BP decoders.In this paper, we propose a new decoder, called the Multiple-Bases Belief-Propagation List Decoder (MBBP-LD), for Quantum Low-Density Parity-Check (QLDPC) codes. It extends the Multiple-Bases Belief-Propagation (MBBP) framework, originally developed for classical cyclic LDPC codes. The proposed method preserves the linear-time complexity of standard BP decoder while improving the logical error rate. To further reduce the logical error rate, a new decision rule is introduced for the post-processing list decoder, outperforming the conventional least-metric selector (LMS) criterion. For the recently developed and implemented bivariate bicycle (BB) code with parameters $[[144,12,12]]$, our proposed MBBP-LD decoder achieves up to 40\% lower logical error rate compared to the state-of-the-art decoder for short QLDPC codes, i.e., BP with ordered-statistics decoding (BP-OSD), while retaining the linear-time complexity of the plain BP decoder. In addition, we explore a new subclass of BB codes, that we refer to as the univariate bicycle (UB) codes, specifically with lower-weight parity checks ($w=6,8$). This reduces the polynomial search space for the code compared to general BB codes, i.e., by reducing the search space over two polynomial components in BB codes to just a single polynomial component in UB codes. Simulations demonstrate the promising performance of these codes under various types of BP decoders.

arXiv:2511.02951Nov 4, 20250 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.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:2609.06572Sep 8, 20260 citations

Spectral Theory of Semisimple Bivariate Bicycle Codes

Eric Sabo, Mahir Bilen Can, David Marquis

Extending the classical theory of two-dimensional cyclic codes, we develop an algebraic approach to bivariate bicycle codes. Using Frobenius-orbit idempotents, formulas for logical dimensions are derived and lower bounds on minimum distances are established. A systematic theory of code symmetries is formulated to construct a structured block-monomial subgroup of coordinate permutations. Several explicit examples show how to generate these codes from first principles without relying on numerical searches. An appendix extends the analysis to BCH-based product constructions.Extending the classical theory of two-dimensional cyclic codes, we develop an algebraic approach to bivariate bicycle codes. Using Frobenius-orbit idempotents, formulas for logical dimensions are derived and lower bounds on minimum distances are established. A systematic theory of code symmetries is formulated to construct a structured block-monomial subgroup of coordinate permutations. Several explicit examples show how to generate these codes from first principles without relying on numerical searches. An appendix extends the analysis to BCH-based product constructions.

quant-pharXiv:2608.27565Aug 31, 2026

Distance Bounds for Generalized Bicycle Codes

Renyu Wang, Leonid P. Pryadko

Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ansätze, unrestricted GB codes may have linear distance scaling. In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement errors. For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in D≤w−1 dimensions, and lower existence bounds which give d≥O(n1/2). We have also conducted an exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8; the observed distance scaling is consistent with A(w)n1/2+B(w), where n is the code length and A(w) is increasing with w.Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ansätze, unrestricted GB codes may have linear distance scaling. In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement errors. For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in D≤w−1 dimensions, and lower existence bounds which give d≥O(n1/2). We have also conducted an exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8; the observed distance scaling is consistent with A(w)n1/2+B(w), where n is the code length and A(w) is increasing with w.

arXiv:2203.17216SymmetryJun 30, 202215 citations

10^3

) demonstrate competitive performance relative to existing GB and bivariate bicycle (BB) codes despite the additional algebraic restriction.

Z

-logical quotient

K/S

over

\mathbb F_2[x,y]/(x^\ell-1,y^m-1)

and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity

d_Z=\min(d_{\mathcal A},d_{\mathcal C})

. Using the Frobenius structure of the finite group algebra, we prove

r_{\mathcal A}=r_{\mathcal C}=k/2

for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances

4,6,10,10,12,18

of the six standard BB codes of lengths

18

288

and enumerates every minimum-weight logical operator. The resulting census shows that

[[108,8,10]]

is the only one of the six whose distance is attained in a single component, with

d_{\mathcal C}=10

and

d_{\mathcal A}=12

Quantum Bicycle LDPC Codes with High

kd^2/n

from Divisor-Driven Search

[[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.

[[144,12,12]]

, our proposed MBBP-LD decoder achieves up to 40\% lower logical error rate compared to the state-of-the-art decoder for short QLDPC codes, i.e., BP with ordered-statistics decoding (BP-OSD), while retaining the linear-time complexity of the plain BP decoder. In addition, we explore a new subclass of BB codes, that we refer to as the univariate bicycle (UB) codes, specifically with lower-weight parity checks (

w=6,8

). This reduces the polynomial search space for the code compared to general BB codes, i.e., by reducing the search space over two polynomial components in BB codes to just a single polynomial component in UB codes. Simulations demonstrate the promising performance of these codes under various types of BP decoders.

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.