arXiv:2609.39395

Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes

Mohammad Rowshan, Simon Devitt

quant-phcs.ITSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations

Abstract

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 $d_{\mathcal A}=12$.

Comments: 15 pages,6 figures, 9 tables

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arXiv:2609.39395

Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes

Mohammad Rowshan, Simon Devitt

quant-phcs.ITSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations

arXiv HTML PDF

Abstract

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

Comments: 15 pages,6 figures, 9 tables

Related papers

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.

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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.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

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Mohammad Rowshan

Faulty syndrome measurements can corrupt an otherwise correct quantum-error correction step. Bivariate bicycle (BB) codes contain dependent stabilizer checks, so every valid syndrome obeys additional parity constraints, or metachecks. We study how far this built-in redundancy can identify measurement faults and when the remaining ambiguity is unavoidable, while separately checking the code's logical structure. A logical decomposition is used as a preliminary safety check: it identifies a $k/2$ -dimensional annihilator subspace and a $k/2$ -dimensional colon quotient, and the minimum-weight logical need not be visible from the annihilator side alone. On the measurement side, translation symmetry partitions syndrome locations into classes that carry identical metacheck information. This gives an exact characterization of the leading single-fault ambiguity, a bound on how many fault locations can be distinguished, and a family-level repair limit when the encoded dimension stays bounded while the block length grows. Under a static-data assumption, the same calculation gives the minimum number of checks that must be remeasured to remove every single-fault ambiguity. Exact finite-code calculations illustrate both regimes: all single measurement faults are distinguishable in a 72-qubit BB code, whereas the 144-qubit Gross code merges its 72 syndrome locations into 36 indistinguishable pairs. In a 108-qubit example, one logical component first appears at weight 12 while the other contains a weight-10 logical. Sustained phenomenological experiments show that joint data--measurement decoding is more robust than a separated repair stage on the more ambiguous codes. The resulting tests apply to general two-block BB codes, including non-coprime periods and repeated-root cases.Faulty syndrome measurements can corrupt an otherwise correct quantum-error correction step. Bivariate bicycle (BB) codes contain dependent stabilizer checks, so every valid syndrome obeys additional parity constraints, or metachecks. We study how far this built-in redundancy can identify measurement faults and when the remaining ambiguity is unavoidable, while separately checking the code's logical structure. A logical decomposition is used as a preliminary safety check: it identifies a $k/2$-dimensional annihilator subspace and a $k/2$-dimensional colon quotient, and the minimum-weight logical need not be visible from the annihilator side alone. On the measurement side, translation symmetry partitions syndrome locations into classes that carry identical metacheck information. This gives an exact characterization of the leading single-fault ambiguity, a bound on how many fault locations can be distinguished, and a family-level repair limit when the encoded dimension stays bounded while the block length grows. Under a static-data assumption, the same calculation gives the minimum number of checks that must be remeasured to remove every single-fault ambiguity. Exact finite-code calculations illustrate both regimes: all single measurement faults are distinguishable in a 72-qubit BB code, whereas the 144-qubit Gross code merges its 72 syndrome locations into 36 indistinguishable pairs. In a 108-qubit example, one logical component first appears at weight 12 while the other contains a weight-10 logical. Sustained phenomenological experiments show that joint data--measurement decoding is more robust than a separated repair stage on the more ambiguous codes. The resulting tests apply to general two-block BB codes, including non-coprime periods and repeated-root cases.

quant-pharXiv:2609.39121Oct 1, 20260 citations

Single-Shot and Few-Shot Decoding via Stabilizer Redundancy in Bivariate Bicycle Codes

Mohammad Rowshan

Bivariate bicycle (BB) codes are a prominent class of quantum LDPC codes constructed from group algebras. While the logical dimension and quantum distance of \emph{coprime} BB codes are known to be determined by a greatest common divisor polynomial $g(z)$ , the properties governing their fault tolerance under noisy measurement have remained implicit. In this work, we prove that this same polynomial $g(z)$ dictates the code's stabilizer redundancy and the structure of the classical \emph{syndrome codes} required for single-shot decoding. We derive a strict equality between the quantum rate and the stabilizer redundancy density, and we provide BCH-like bounds on the achievable single-shot measurement error tolerance. Guided by this framework, we construct small coprime BB codes with significantly improved syndrome distance ( $d_S$ ) and evaluate them using BP+OSD. Our analysis reveals a structural bottleneck: within the coprime BB ansatz, high quantum rate imposes an upper bound on syndrome distance, limiting single-shot performance. These results provide concrete algebraic design rules for next-generation 2BGA codes in measurement-limited architectures.Bivariate bicycle (BB) codes are a prominent class of quantum LDPC codes constructed from group algebras. While the logical dimension and quantum distance of \emph{coprime} BB codes are known to be determined by a greatest common divisor polynomial $g(z)$, the properties governing their fault tolerance under noisy measurement have remained implicit. In this work, we prove that this same polynomial $g(z)$ dictates the code's stabilizer redundancy and the structure of the classical \emph{syndrome codes} required for single-shot decoding. We derive a strict equality between the quantum rate and the stabilizer redundancy density, and we provide BCH-like bounds on the achievable single-shot measurement error tolerance. Guided by this framework, we construct small coprime BB codes with significantly improved syndrome distance ($d_S$) and evaluate them using BP+OSD. Our analysis reveals a structural bottleneck: within the coprime BB ansatz, high quantum rate imposes an upper bound on syndrome distance, limiting single-shot performance. These results provide concrete algebraic design rules for next-generation 2BGA codes in measurement-limited architectures.

cs.ITCross-listedarXiv:2601.01137IEEE International Symposium on Information Theory (ISIT)Jan 6, 20260 citations

Exact distances for 1027 bivariate-bicycle codes and a coset upper bound

Advatiya Thalluri

A bivariate-bicycle (BB) code is defined by two polynomials A, B over F2[x,y]/(x^l - 1, y^m - 1) and has n = 2lm qubits. We fix the polynomials in two families and vary only (l, m) with n <= 400, and report the exact distance of 1027 codes. We then give an upper bound on the distance, built from logicals confined to unions of shifted cosets of a cyclic subgroup H in each block (r <= 2 for every H, and r = 3 when the number of cosets is at most 12), and compare it with the data. It equals the exact distance on 1014 of the 1027 codes (an in-sample fit that is not blind). Two blind tests were run with the predictor and predictions hashed before any distance was read. On 25 distinct catalogued codes it is tight on 24 (96%). On 150 random weight-6 BB codes it is tight on only 102 (68%, 95% confidence interval 0.599-0.754), which fails the preregistered criterion of at least 90%. The bound is never below the exact distance in either test. The bound is therefore tight on 24 of 25 catalogued codes but not on random codes, and the empirical conjecture of Section 5 is restricted to the two families studied.A bivariate-bicycle (BB) code is defined by two polynomials A, B over F2[x,y]/(x^l - 1, y^m - 1) and has n = 2lm qubits. We fix the polynomials in two families and vary only (l, m) with n <= 400, and report the exact distance of 1027 codes. We then give an upper bound on the distance, built from logicals confined to unions of shifted cosets of a cyclic subgroup H in each block (r <= 2 for every H, and r = 3 when the number of cosets is at most 12), and compare it with the data. It equals the exact distance on 1014 of the 1027 codes (an in-sample fit that is not blind). Two blind tests were run with the predictor and predictions hashed before any distance was read. On 25 distinct catalogued codes it is tight on 24 (96%). On 150 random weight-6 BB codes it is tight on only 102 (68%, 95% confidence interval 0.599-0.754), which fails the preregistered criterion of at least 90%. The bound is never below the exact distance in either test. The bound is therefore tight on 24 of 25 catalogued codes but not on random codes, and the empirical conjecture of Section 5 is restricted to the two families studied.

Oct 28, 20260 citations

Z

-logical quotient

K/S

over

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

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

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

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.

10^3

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

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.

k/2

-dimensional annihilator subspace and a

k/2

-dimensional colon quotient, and the minimum-weight logical need not be visible from the annihilator side alone. On the measurement side, translation symmetry partitions syndrome locations into classes that carry identical metacheck information. This gives an exact characterization of the leading single-fault ambiguity, a bound on how many fault locations can be distinguished, and a family-level repair limit when the encoded dimension stays bounded while the block length grows. Under a static-data assumption, the same calculation gives the minimum number of checks that must be remeasured to remove every single-fault ambiguity. Exact finite-code calculations illustrate both regimes: all single measurement faults are distinguishable in a 72-qubit BB code, whereas the 144-qubit Gross code merges its 72 syndrome locations into 36 indistinguishable pairs. In a 108-qubit example, one logical component first appears at weight 12 while the other contains a weight-10 logical. Sustained phenomenological experiments show that joint data--measurement decoding is more robust than a separated repair stage on the more ambiguous codes. The resulting tests apply to general two-block BB codes, including non-coprime periods and repeated-root cases.

g(z)

, the properties governing their fault tolerance under noisy measurement have remained implicit. In this work, we prove that this same polynomial

g(z)

dictates the code's stabilizer redundancy and the structure of the classical \emph{syndrome codes} required for single-shot decoding. We derive a strict equality between the quantum rate and the stabilizer redundancy density, and we provide BCH-like bounds on the achievable single-shot measurement error tolerance. Guided by this framework, we construct small coprime BB codes with significantly improved syndrome distance (

d_S

) and evaluate them using BP+OSD. Our analysis reveals a structural bottleneck: within the coprime BB ansatz, high quantum rate imposes an upper bound on syndrome distance, limiting single-shot performance. These results provide concrete algebraic design rules for next-generation 2BGA codes in measurement-limited architectures.