[[88,6,6]] · QEC Challenge

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[[88,6,6]] d =

kd²/n

2.455

X/Z

0.0303

3.0

layers

swaps

199

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Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 6 · witness weight 6 (claimed exact)

witness operator (support, 6 qubits)

[2, 42, 50, 58, 76, 86]

d_Z 6 · witness weight 6 (claimed exact)

witness operator (support, 6 qubits)

[1, 17, 19, 27, 37, 87]

certificate exact, d = 6 · scipy/HiGHS cutoff IP

X: no logical < 6 exists; Z: no logical < 6 exists

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 2–6 (mean 4.714) · H_Z 2–6 (mean 4.674)

qubit degrees H_X 1–4 (mean 2.25) · H_Z 1–4 (mean 2.284)

trapping sets H_X (1,1)×23 (2,0)×4 (3,0)×11 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,1): 23 (1,2): 24 (1,3): 37 (1,4): 4 (2,0): 4 (2,1): 29 (2,2): 81 (2,3): 106 (2,4): 168 (2,5): 19 (2,6): 2 (3,0): 11 (3,1): 61 (3,2): 204 (3,3): 559 (3,4): 831 (3,5): 974 (3,6): 209 (3,7): 170 (3,8): 7 (3,9): 4

trapping sets H_Z (1,1)×22 (2,0)×6 (3,0)×6 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 22 (1,2): 24 (1,3): 37 (1,4): 5 (2,0): 6 (2,1): 31 (2,2): 68 (2,3): 126 (2,4): 149 (2,5): 35 (2,6): 3 (3,0): 6 (3,1): 39 (3,2): 253 (3,3): 578 (3,4): 871 (3,5): 819 (3,6): 349 (3,7): 169 (3,8): 33 (3,9): 6

witness diameter X 5.9218 · Z 7.3254 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

X checkZ checkqubit site (53)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

routing cost 199 nearest-neighbor SWAPs per round in total, at most 3 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors Liang, Zijian and Eberhardt, Jens Niklas and Chen, Yu-An

provenance literature baseline

construction Liang-Eberhardt-Chen k=6 family f=1+x+x^-1y^-2, g=1+y+xy^-1; (6,8) open-boundary lattice.

model classical construction (no AI model)

reference arXiv:2504.08887

date 2025-04-11

notes Reference baseline (Liang, Eberhardt, Chen). Bilayer grid layout added 2026-07-23, recovered from the check structure by template matching: the 18+24 weight-6 bulk checks are translated copies of (f, g), which pins 84 of 88 qubits on the two-layer grid; the 4 notch-boundary qubits were placed by exhaustive local search. Measured interaction radius sqrt(10). Layout shear-optimized 2026-07-23: linear map [[1.000000,-0.318933],[0.000000,0.968940]] applied to the template-solved grid with the 4 notch-boundary qubits re-placed in the sheared lattice, reducing the interaction radius sqrt(10) -> 3.0; same code, same check matrices.

family bivariate bicycle (a tag, not a ranking)

locality 2D-local bilayer (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 42 (max weight 6) · Z-checks 43 (max weight 6)

H_X (42 checks, sparse supports)

[0, 10, 18, 49, 50, 56] [1, 11, 19, 50, 51, 57] [2, 12, 20, 51, 52, 58] [3, 13, 21, 52, 53, 59] [4, 14, 22, 53, 54, 60] [5, 15, 23, 54, 55, 61] [8, 18, 26, 57, 58, 64] [9, 19, 27, 58, 59, 65] [10, 20, 28, 59, 60, 66] [11, 21, 29, 60, 61, 67] [12, 22, 30, 61, 62, 68] [13, 23, 31, 62, 63, 69] [16, 26, 34, 65, 66, 72] [17, 27, 35, 66, 67, 73] [18, 28, 36, 67, 68, 74] [19, 29, 37, 68, 69, 75] [20, 30, 38, 69, 70, 76] [21, 31, 39, 70, 71, 77] [2, 40] [3, 41] [4, 42] [5, 43] [6, 44] [7, 45] [32, 81, 82] [33, 82, 83] [34, 83, 84] [35, 84, 85] [36, 85, 86] [37, 86, 87] [10, 40, 41, 42, 48] [11, 41, 42, 43, 49] [12, 42, 43, 44, 50] [13, 43, 44, 45, 51] [14, 44, 45, 46, 52] [15, 45, 46, 47, 53] [24, 34, 73, 74, 80] [25, 35, 74, 75, 81] [26, 36, 75, 76, 82] [27, 37, 76, 77, 83] [28, 38, 77, 78, 84] [29, 39, 78, 79, 85]

H_Z (43 checks, sparse supports)

[2, 8, 9, 40, 48, 58] [3, 9, 10, 41, 49, 59] [4, 10, 11, 42, 50, 60] [5, 11, 12, 43, 51, 61] [6, 12, 13, 44, 52, 62] [7, 13, 14, 45, 53, 63] [10, 16, 17, 48, 56, 66] [11, 17, 18, 49, 57, 67] [12, 18, 19, 50, 58, 68] [13, 19, 20, 51, 59, 69] [14, 20, 21, 52, 60, 70] [15, 21, 22, 53, 61, 71] [18, 24, 25, 56, 64, 74] [19, 25, 26, 57, 65, 75] [20, 26, 27, 58, 66, 76] [21, 27, 28, 59, 67, 77] [22, 28, 29, 60, 68, 78] [23, 29, 30, 61, 69, 79] [26, 32, 33, 64, 72, 82] [27, 33, 34, 65, 73, 83] [28, 34, 35, 66, 74, 84] [29, 35, 36, 67, 75, 85] [30, 36, 37, 68, 76, 86] [31, 37, 38, 69, 77, 87] [0, 56] [8, 64] [16, 72] [24, 80] [1, 57, 64] [9, 65, 72] [17, 73, 80] [1, 8, 57] [9, 16, 65] [17, 24, 73] [25, 32, 81] [15, 47, 55] [23, 55, 63] [31, 63, 71] [39, 71, 79] [14, 15, 46, 54] [22, 23, 54, 62] [30, 31, 62, 70] [38, 39, 70, 78]

Code ID 88-6-6 · download JSON · raw on GitHub