← back to the stabilizer board
[[180,2,11]] d ≤stabilizer
n
180
k
2
d
11
kd²/n
1.344
w
5

Share this result

Distance

a general stabilizer code has no X and Z sides: d is the minimum Pauli weight of a nontrivial logical operator (a Y factor counts one qubit), and the witness is one Pauli operator that commutes with every generator and is not a product of them
d 11 · witness Pauli weight 11 (claimed upper_bound)
witness operator (Pauli string, 11 qubits)
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIIIIIZIIIIIIIIIZIIIIZIIIZIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIZIIIZIIIIIIIIIIIIIIII X: [] Z: [30, 40, 83, 97, 107, 112, 116, 144, 154, 159, 163]
certificate none yet · distance stands as a self-certified upper bound (d ≤); the Pauli-weight certifier is not built yet, so stabilizer entries cannot earn d= for now

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth S 4 (shortest cycle of the generator Tanner graph; longer is friendlier to belief propagation)
check weights S 5
qubit degrees S 5
trapping sets S (1,5)×180 (2,6)×720 (3,7)×2520 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for S
(1,5): 180 (2,6): 720 (2,8): 360 (3,7): 2520 (3,9): 4680 (3,11): 1080

Construction & provenance

authors @cbjuan
provenance submitted through the challenge
novelty novelty not audited
construction Perturbed bivariate bicycle (PBB) code: ell=15, m=6; A=1+x2y2, B=1+x2y3+x4, C-perturbation=1+x2y2, D-perturbation=1+x4 (0-indexed exponent pairs: A=[[0,0],[2,2]], B=[[0,0],[2,3],[4,0]], C=[[0,0],[2,2]], D=[[0,0],[4,0]]); match W5PBB-180-2-11-8ee754a5, a standalone explicit presentation at this native (ell,m) certified connected and exactly distance-11 by MILP transfer from an isomorphic-component parent (qcode-discovery @ d6abdbd).
model Claude Claude Opus 5, GPT-5.6 Sol (claimed, not verified)
date 2026-10-07
notes Checked for duplication against the current board: no existing entry (CSS or stabilizer) at this [[n,k,d]]. First published in arXiv:2610.06623.
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[180,2,11]] — exact non-CSS perturbed bivariate bicycle (PBB) code

Paper: arXiv:2610.06623, "Large Language Model-Guided Discovery of Weight-Five Bivariate Bicycle Codes" (match W5PBB-180-2-11-8ee754a5).

Direction & hypothesis

Target: the general-stabilizer board's unrestricted x weight-6 (and looser) cells. Same PBB construction family as the companion submissions in this batch (block-1 z-part [C | D] added to a CSS bivariate-bicycle stabilizer matrix: H = (A B | C D; 0 0 | B^T A^T)).

What was searched

Source: a weight-5-targeted LLM-guided evolutionary search (OpenEvolve, "pbb-large" campaign) over 4-tuples (A, B, C, D), screened by k, isotropy, the combined non-CSS gate (evaluation/noncss_gate.py), and a BP-OSD distance estimate. The underlying code was first found as one of 2 isomorphic components of a disconnected parent presentation at (ell=15, m=12), n=360, k=4 (evaluation/connectivity.py:stabilizer_components + components_isomorphic); the parent's exact MILP distance transfers to each component. A standalone, explicit presentation of one component at its native (ell=15, m=6) was then independently constructed and certified (results/weight5_pbb_component_certifications.jsonl, match W5PBB-180-2-11-8ee754a5): re-run through the same non-CSS screens and confirmed connected.

Evidence trail

(ell, m) = (15, 6), A = 1 + x^2y^2, B = 1 + x^2y^3 + x^4, perturbation C = 1 + x^2y^2, D = 1 + x^4 (0-indexed exponent pairs: A=[[0,0],[2,2]], B=[[0,0],[2,3],[4,0]], C=[[0,0],[2,2]], D=[[0,0],[4,0]]). n = 2·15·6 = 180, k = 2, max check weight 5.

  • Source repo's distance ladder: deep per-logical MILP proved all logicals

    • optimal at weight 11 on the (15,12) parent -- exact, transferred to this component by the isomorphism/homogeneity argument.

  • Independent re-verification in *this* repository: rebuilt the symplectic

    • check matrix directly from the (15,6) component's own generators via evaluation/pbb_code.py:build_pbb_code, then ran this repo's own cli/qldpc.py submit --dry-run (RIS search, 20,000 trials, gf2_fast accelerator at 1,000,000 trials). Independently found d<=11, matching the source's exact claim.

Final claim here: witness-backed upper_bound; the source repository's own evidence tier is exact.

Dead ends

Not applicable -- this submission mined an existing, already-vetted and now-published campaign result.

Tools

Model: jointly attributed to Claude Opus 5 and GPT-5.6 Sol (both aliases recovered against the parent's lineage log, 23 observations), matches provenance.model. This submission's rebuild, independent re-verification, and packaging in *this* repository was done separately, using only this repository's own tooling.

Reproduction

from evaluation.pbb_code import build_pbb_code
code = build_pbb_code(ell=15, m=6,
                       A_terms=[(0,0),(2,2)],
                       B_terms=[(0,0),(2,3),(4,0)],
                       C_terms=[(0,0),(2,2)],
                       D_terms=[(0,0),(4,0)])
# code.matrix is the (180, 360) symplectic stabilizer matrix [X-part | Z-part]

Source: github.com/qiskit-community/qcode-discovery @ d6abdbd, evaluation/pbb_code.py, results/weight5_pbb_component_certifications.jsonl.

Stabilizer generators

generators 180 (max weight 5; 90 mixed X/Z) (one binary symplectic matrix S = (A | B); generator i is X on A_i and Z on B_i, Y where both)
generators (180, Pauli strings on 180 qubits)
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symplectic rows (A | B) (180, sparse supports)
X: [0, 14, 90, 105, 114] Z: [0, 14, 90, 114] X: [1, 15, 91, 106, 115] Z: [1, 15, 91, 115] X: [2, 16, 92, 107, 116] Z: [2, 16, 92, 116] X: [3, 17, 93, 102, 117] Z: [3, 17, 93, 117] X: [4, 12, 94, 103, 118] Z: [4, 12, 94, 118] X: [5, 13, 95, 104, 119] Z: [5, 13, 95, 119] X: [6, 20, 96, 111, 120] Z: [6, 20, 96, 120] X: [7, 21, 97, 112, 121] Z: [7, 21, 97, 121] X: [8, 22, 98, 113, 122] Z: [8, 22, 98, 122] X: [9, 23, 99, 108, 123] Z: [9, 23, 99, 123] X: [10, 18, 100, 109, 124] Z: [10, 18, 100, 124] X: [11, 19, 101, 110, 125] Z: [11, 19, 101, 125] X: [12, 26, 102, 117, 126] Z: [12, 26, 102, 126] X: [13, 27, 103, 118, 127] Z: [13, 27, 103, 127] X: [14, 28, 104, 119, 128] Z: [14, 28, 104, 128] X: [15, 29, 105, 114, 129] Z: [15, 29, 105, 129] X: [16, 24, 106, 115, 130] Z: [16, 24, 106, 130] X: [17, 25, 107, 116, 131] Z: [17, 25, 107, 131] X: [18, 32, 108, 123, 132] Z: [18, 32, 108, 132] X: [19, 33, 109, 124, 133] Z: [19, 33, 109, 133] X: [20, 34, 110, 125, 134] Z: [20, 34, 110, 134] X: [21, 35, 111, 120, 135] Z: [21, 35, 111, 135] X: [22, 30, 112, 121, 136] Z: [22, 30, 112, 136] X: [23, 31, 113, 122, 137] Z: [23, 31, 113, 137] X: [24, 38, 114, 129, 138] Z: [24, 38, 114, 138] X: [25, 39, 115, 130, 139] Z: [25, 39, 115, 139] X: [26, 40, 116, 131, 140] Z: [26, 40, 116, 140] X: [27, 41, 117, 126, 141] Z: [27, 41, 117, 141] X: [28, 36, 118, 127, 142] Z: [28, 36, 118, 142] X: [29, 37, 119, 128, 143] Z: [29, 37, 119, 143] X: [30, 44, 120, 135, 144] Z: [30, 44, 120, 144] X: [31, 45, 121, 136, 145] Z: [31, 45, 121, 145] X: [32, 46, 122, 137, 146] Z: [32, 46, 122, 146] X: [33, 47, 123, 132, 147] Z: [33, 47, 123, 147] X: [34, 42, 124, 133, 148] Z: [34, 42, 124, 148] X: [35, 43, 125, 134, 149] Z: [35, 43, 125, 149] X: [36, 50, 126, 141, 150] Z: [36, 50, 126, 150] X: [37, 51, 127, 142, 151] Z: [37, 51, 127, 151] X: [38, 52, 128, 143, 152] Z: [38, 52, 128, 152] X: [39, 53, 129, 138, 153] Z: [39, 53, 129, 153] X: [40, 48, 130, 139, 154] Z: [40, 48, 130, 154] X: [41, 49, 131, 140, 155] Z: [41, 49, 131, 155] X: [42, 56, 132, 147, 156] Z: [42, 56, 132, 156] X: [43, 57, 133, 148, 157] Z: [43, 57, 133, 157] X: [44, 58, 134, 149, 158] Z: [44, 58, 134, 158] X: [45, 59, 135, 144, 159] Z: [45, 59, 135, 159] X: [46, 54, 136, 145, 160] Z: [46, 54, 136, 160] X: [47, 55, 137, 146, 161] Z: [47, 55, 137, 161] X: [48, 62, 138, 153, 162] Z: [48, 62, 138, 162] X: [49, 63, 139, 154, 163] Z: [49, 63, 139, 163] X: [50, 64, 140, 155, 164] Z: [50, 64, 140, 164] X: [51, 65, 141, 150, 165] Z: [51, 65, 141, 165] X: [52, 60, 142, 151, 166] Z: [52, 60, 142, 166] X: [53, 61, 143, 152, 167] Z: [53, 61, 143, 167] X: [54, 68, 144, 159, 168] Z: [54, 68, 144, 168] X: [55, 69, 145, 160, 169] Z: [55, 69, 145, 169] X: [56, 70, 146, 161, 170] Z: [56, 70, 146, 170] X: [57, 71, 147, 156, 171] Z: [57, 71, 147, 171] X: [58, 66, 148, 157, 172] Z: [58, 66, 148, 172] X: [59, 67, 149, 158, 173] Z: [59, 67, 149, 173] X: [60, 74, 150, 165, 174] Z: [60, 74, 150, 174] X: [61, 75, 151, 166, 175] Z: [61, 75, 151, 175] X: [62, 76, 152, 167, 176] Z: [62, 76, 152, 176] X: [63, 77, 153, 162, 177] Z: [63, 77, 153, 177] X: [64, 72, 154, 163, 178] Z: [64, 72, 154, 178] X: [65, 73, 155, 164, 179] Z: [65, 73, 155, 179] X: [66, 80, 90, 156, 171] Z: [66, 80, 90, 156] X: [67, 81, 91, 157, 172] Z: [67, 81, 91, 157] X: [68, 82, 92, 158, 173] Z: [68, 82, 92, 158] X: [69, 83, 93, 159, 168] Z: [69, 83, 93, 159] X: [70, 78, 94, 160, 169] Z: [70, 78, 94, 160] X: [71, 79, 95, 161, 170] Z: [71, 79, 95, 161] X: [72, 86, 96, 162, 177] Z: [72, 86, 96, 162] X: [73, 87, 97, 163, 178] Z: [73, 87, 97, 163] X: [74, 88, 98, 164, 179] Z: [74, 88, 98, 164] X: [75, 89, 99, 165, 174] Z: [75, 89, 99, 165] X: [76, 84, 100, 166, 175] Z: [76, 84, 100, 166] X: [77, 85, 101, 167, 176] Z: [77, 85, 101, 167] X: [2, 78, 93, 102, 168] Z: [2, 78, 102, 168] X: [3, 79, 94, 103, 169] Z: [3, 79, 103, 169] X: [4, 80, 95, 104, 170] Z: [4, 80, 104, 170] X: [5, 81, 90, 105, 171] Z: [5, 81, 105, 171] X: [0, 82, 91, 106, 172] Z: [0, 82, 106, 172] X: [1, 83, 92, 107, 173] Z: [1, 83, 107, 173] X: [8, 84, 99, 108, 174] Z: [8, 84, 108, 174] X: [9, 85, 100, 109, 175] Z: [9, 85, 109, 175] X: [10, 86, 101, 110, 176] Z: [10, 86, 110, 176] X: [11, 87, 96, 111, 177] Z: [11, 87, 111, 177] X: [6, 88, 97, 112, 178] Z: [6, 88, 112, 178] X: [7, 89, 98, 113, 179] Z: [7, 89, 113, 179] X: [] Z: [0, 66, 81, 90, 172] X: [] Z: [1, 67, 82, 91, 173] X: [] Z: [2, 68, 83, 92, 168] X: [] Z: [3, 69, 78, 93, 169] X: [] Z: [4, 70, 79, 94, 170] X: [] Z: [5, 71, 80, 95, 171] X: [] Z: [6, 72, 87, 96, 178] X: [] Z: [7, 73, 88, 97, 179] X: [] Z: [8, 74, 89, 98, 174] X: [] Z: [9, 75, 84, 99, 175] X: [] Z: [10, 76, 85, 100, 176] X: [] Z: [11, 77, 86, 101, 177] X: [] Z: [3, 12, 78, 94, 102] X: [] Z: [4, 13, 79, 95, 103] X: [] Z: [5, 14, 80, 90, 104] X: [] Z: [0, 15, 81, 91, 105] X: [] Z: [1, 16, 82, 92, 106] X: [] Z: [2, 17, 83, 93, 107] X: [] Z: [9, 18, 84, 100, 108] X: [] Z: [10, 19, 85, 101, 109] X: [] Z: [11, 20, 86, 96, 110] X: [] Z: [6, 21, 87, 97, 111] X: [] Z: [7, 22, 88, 98, 112] X: [] Z: [8, 23, 89, 99, 113] X: [] Z: [0, 15, 24, 106, 114] X: [] Z: [1, 16, 25, 107, 115] X: [] Z: [2, 17, 26, 102, 116] X: [] Z: [3, 12, 27, 103, 117] X: [] Z: [4, 13, 28, 104, 118] X: [] Z: [5, 14, 29, 105, 119] X: [] Z: [6, 21, 30, 112, 120] X: [] Z: [7, 22, 31, 113, 121] X: [] Z: [8, 23, 32, 108, 122] X: [] Z: [9, 18, 33, 109, 123] X: [] Z: [10, 19, 34, 110, 124] X: [] Z: [11, 20, 35, 111, 125] X: [] Z: [12, 27, 36, 118, 126] X: [] Z: [13, 28, 37, 119, 127] X: [] Z: [14, 29, 38, 114, 128] X: [] Z: [15, 24, 39, 115, 129] X: [] Z: [16, 25, 40, 116, 130] X: [] Z: [17, 26, 41, 117, 131] X: [] Z: [18, 33, 42, 124, 132] X: [] Z: [19, 34, 43, 125, 133] X: [] Z: [20, 35, 44, 120, 134] X: [] Z: [21, 30, 45, 121, 135] X: [] Z: [22, 31, 46, 122, 136] X: [] Z: [23, 32, 47, 123, 137] X: [] Z: [24, 39, 48, 130, 138] X: [] Z: [25, 40, 49, 131, 139] X: [] Z: [26, 41, 50, 126, 140] X: [] Z: [27, 36, 51, 127, 141] X: [] Z: [28, 37, 52, 128, 142] X: [] Z: [29, 38, 53, 129, 143] X: [] Z: [30, 45, 54, 136, 144] X: [] Z: [31, 46, 55, 137, 145] X: [] Z: [32, 47, 56, 132, 146] X: [] Z: [33, 42, 57, 133, 147] X: [] Z: [34, 43, 58, 134, 148] X: [] Z: [35, 44, 59, 135, 149] X: [] Z: [36, 51, 60, 142, 150] X: [] Z: [37, 52, 61, 143, 151] X: [] Z: [38, 53, 62, 138, 152] X: [] Z: [39, 48, 63, 139, 153] X: [] Z: [40, 49, 64, 140, 154] X: [] Z: [41, 50, 65, 141, 155] X: [] Z: [42, 57, 66, 148, 156] X: [] Z: [43, 58, 67, 149, 157] X: [] Z: [44, 59, 68, 144, 158] X: [] Z: [45, 54, 69, 145, 159] X: [] Z: [46, 55, 70, 146, 160] X: [] Z: [47, 56, 71, 147, 161] X: [] Z: [48, 63, 72, 154, 162] X: [] Z: [49, 64, 73, 155, 163] X: [] Z: [50, 65, 74, 150, 164] X: [] Z: [51, 60, 75, 151, 165] X: [] Z: [52, 61, 76, 152, 166] X: [] Z: [53, 62, 77, 153, 167] X: [] Z: [54, 69, 78, 160, 168] X: [] Z: [55, 70, 79, 161, 169] X: [] Z: [56, 71, 80, 156, 170] X: [] Z: [57, 66, 81, 157, 171] X: [] Z: [58, 67, 82, 158, 172] X: [] Z: [59, 68, 83, 159, 173] X: [] Z: [60, 75, 84, 166, 174] X: [] Z: [61, 76, 85, 167, 175] X: [] Z: [62, 77, 86, 162, 176] X: [] Z: [63, 72, 87, 163, 177] X: [] Z: [64, 73, 88, 164, 178] X: [] Z: [65, 74, 89, 165, 179]
Code ID 180-2-11 · download JSON · raw on GitHub