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

• URL: http://arxiv.org/abs/2609.06572v1
• Date: Sun, 06 Sep 2026 12:34:34 GMT
• Title: Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark
• Abstract summary: BB codes of Bravyi emphet al.citeBravyi2024 are quantum low-density parity-check codes with weight-$6$ checks.<n>We develop the structure theory of BB-type codes with weight-$8$ checks (weight-$4$ generators) and use it to construct and certify new codes.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• Ordered-Angle Coding for Exact Multiuser Unanimity Testing [0.0]
  We introduce ordered-angle coding for binary-unanimity testing among $n$ transformation-only users in a serial quantum architecture.<n>Bell entanglement is not required for the additive algebra, but it keeps the traveler locally maximally mixed in every honest trial.
  arXiv  Detail & Related papers  (2026-09-04T18:28:01Z)
• Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search [3.344730946122235]
  Bicycle quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes.<n>We show that in the case the construction collapses into the ring $F[x]/(xl)$: self-orthogonality is automatic.<n>A computer search based on this framework recovers the short codes.
  arXiv  Detail & Related papers  (2026-08-10T04:45:08Z)
• The Condition-Number Barrier in Sparse Least Squares [77.64108812086542]
  In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in convex optimization cannot be improved by a sparse-time algorithm.<n>We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis.
  arXiv  Detail & Related papers  (2026-08-03T17:57:01Z)
• Near-Optimal Dimension Lower Bounds for Single-Vector Embeddings of Maximum Inner Product Similarity [75.14269295861845]
  For singleton queries, Chamfer becomes maximum inner product similarity (MAX-IP)<n>For every fixed $in(0,1)$, there are constants $A_,c_>0$.<n>Unit-sphere MAX-IP matrix is an exact two-valued affine image of the DNF pattern matrix with gap of at least $8$.
  arXiv  Detail & Related papers  (2026-07-22T17:27:20Z)
• Faster quantum linear system solver beyond the condition number [50.84794327094274]
  We present two quantum algorithms that produce the normalized solution $|xrangle$ of linear system $Ax=| b rangle$ to accuracy $$ with complexity independent of the condition number $=lVert A-1rVert$.<n>Our filtering-based solver is extremely simple with a favorable runtime prefactor.
  arXiv  Detail & Related papers  (2026-07-08T17:49:40Z)
• Rate-2/3 Girth-8 (3,18)-Regular Quantum LDPC Codes from Two-Branch Finite-Field Bases and CPM Lifts [0.0]
  The code is a Calderbank-Shor-Steane (CSS) code with parameters $[34542,23032,dle 310]]$.<n>We do not regard this upper bound as an estimate of the true minimum distance; rather, $dle310$ is the tightest upper bound currently obtained from structural lifts and decoder-produced logical errors.
  arXiv  Detail & Related papers  (2026-06-25T15:08:16Z)
• Breaking the bicycle frame: Coset-based quantum LDPC codes [65.96089068176079]
  Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes.<n>We identify several new quantum LDPC codes, including weight-6 codes with parameters $[48,8,6]]$, $[96,8,10]]$, and $[224,12,16]]$.<n>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.
  arXiv  Detail & Related papers  (2026-06-15T20:20:57Z)
• High-Girth Regular Quantum LDPC Codes from Affine-Coset Structures [0.0]
  We construct a quantum low-density parity-check code family from a length-$512$ Calderbank-Shor--Steane base matrix pair.<n>The base pair is permutation-equivalent to the known SPC(3) product CSS code.
  arXiv  Detail & Related papers  (2026-04-22T17:59:08Z)
• Independent Trivariate Bicycle Codes [0.0]
  We introduce six independent tridimensional bicycle (ITB) codes.<n>We construct four codes including a $[140,6,14]]$ code with $kd2/n = 8.40$.<n>On the superconducting noise model, the $[140,6,14]] code achieves a per-round per-observable rate of $5.6 times 10-5$ at $p = 0.20%$.
  arXiv  Detail & Related papers  (2026-03-18T13:22:40Z)
• Approximating the operator norm of local Hamiltonians via few quantum states [53.16156504455106]
  Consider a Hermitian operator $A$ acting on a complex Hilbert space of $2n$.<n>We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian.<n>We show that whenever $A$ is $d$-local, textiti.e., $deg(A)le d$, we have the following discretization-type inequality.
  arXiv  Detail & Related papers  (2025-09-15T14:26:11Z)
• Generalized toric codes on twisted tori for quantum error correction [9.623534315687825]
  Kitaev toric code is widely considered one of the leading candidates for error correction in fault-tolerant quantum computation.<n>Direct methods to increase its logical dimensions, such as lattice surgery or introducing punctures, often incur prohibitive overheads.<n>We introduce a ring-theoretic approach for efficiently analyzing topological CSS codes in two dimensions.
  arXiv  Detail & Related papers  (2025-03-05T19:00:05Z)
• Quantum $(r,δ)$-locally recoverable codes [37.306043163932905]
  We introduce the quantum counterpart of those codes by defining quantum $(r,delta)$-locally recoverable codes.<n>We show that there is an equivalence between the classical and quantum concepts of $(r,delta)$-local recoverability.
  arXiv  Detail & Related papers  (2024-12-21T11:45:32Z)
• Solving Quadratic Systems with Full-Rank Matrices Using Sparse or Generative Priors [33.0212223058894]
  The problem of recovering a signal from a quadratic system $y_i=boldsymbol xtopboldsymbol A_iboldsymbol x, i=1,ldots,m$ with full-rank matrices $boldsymbol A_i$ frequently arises in applications such as unassigned distance geometry and sub-wavelength imaging. This paper addresses the high-dimensional case where $mll n$ incorporating by prior knowledge of $boldsymbol x$.
  arXiv  Detail & Related papers  (2023-09-16T16:00:07Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.