[[72,6,6]] · QEC Challenge

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[[72,6,6]] d =

kd²/n

3.0

X/Z

0.0104

4.1231

layers

swaps

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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)

[10, 23, 41, 65, 69, 70]

d_Z 6 · witness weight 6 (claimed exact)

witness operator (support, 6 qubits)

[16, 25, 29, 47, 48, 63]

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 4.333) · H_Z 2–6 (mean 4.432)

qubit degrees H_X 1–4 (mean 2.167) · H_Z 1–5 (mean 2.278)

trapping sets H_X (1,1)×19 (2,0)×8 (3,1)×27 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 19 (1,2): 26 (1,3): 23 (1,4): 4 (2,0): 8 (2,1): 24 (2,2): 65 (2,3): 106 (2,4): 77 (2,5): 24 (3,1): 27 (3,2): 255 (3,3): 470 (3,4): 553 (3,5): 421 (3,6): 224 (3,7): 62 (3,8): 17

trapping sets H_Z (1,1)×22 (2,0)×10 (3,1)×24 (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): 19 (1,3): 23 (1,4): 5 (1,5): 3 (2,0): 10 (2,1): 32 (2,2): 45 (2,3): 88 (2,4): 80 (2,5): 33 (2,6): 17 (2,7): 5 (2,8): 1 (3,1): 24 (3,2): 226 (3,3): 371 (3,4): 508 (3,5): 422 (3,6): 300 (3,7): 175 (3,8): 74 (3,9): 17 (3,10): 4

witness diameter X 5.3852 · Z 5.3852 (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 (36)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 96 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 @FarLab and @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction f = 1 + xy + x^-1 y², g = 1 + x^-1 y + x^-2 y^-1; annulus-type boundary engine, (6,6) lattice.

model Claude Opus 4.8 (claimed, not verified)

builds on arXiv:2504.08887

date 2026-06-16

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 36 (max weight 6) · Z-checks 37 (max weight 6)

H_X (36 checks, sparse supports)

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

H_Z (37 checks, sparse supports)

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

Code ID 72-6-6 · download JSON · raw on GitHub