[[60,12,7]] · QEC Challenge

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[[60,12,7]] d =

kd²/n

9.8

X/Z

0.014

7.2801

layers

swaps

521

Share this result

Distance

X/Z asymmetry 1 · d_X = 7, d_Z = 7 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[18, 43, 44, 46, 47, 48, 53]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness operator (support, 7 qubits)

[22, 23, 25, 36, 48, 53, 58]

certificate exact, d = 7 · scipy/HiGHS MILP

X: no logical < 7 exists; Z: no logical < 7 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 8 · H_Z 8

qubit degrees H_X 4 · H_Z 4

trapping sets H_X (1,4)×60 (2,4)×60 (3,4)×210 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 60 (2,4): 60 (2,6): 720 (3,4): 210 (3,6): 2400 (3,8): 9660 (3,10): 1020

trapping sets H_Z (1,4)×60 (2,4)×60 (3,4)×210 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 60 (2,4): 60 (2,6): 720 (3,4): 210 (3,6): 2400 (3,8): 9660 (3,10): 1020

witness diameter X 5.831 · Z 6.3246 (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

layout contributed by @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19

X checkZ checkqubit site (60)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 521 nearest-neighbor SWAPs per round in total, at most 14 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 @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction Coprime BB on Z_5xZ_6, bivariate A=[(0, 0), (1, 1), (2, 2), (2, 1)], B=[(0, 0), (3, 3), (2, 0), (0, 1)]

model MiMo V2.5 (claimed, not verified)

date 2026-09-15

family bivariate bicycle (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 8 (computed)

How this code was found

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

[[60,12,7]] — from arXiv:2408.10001v6

Direction & hypothesis

Reconstructed from arXiv:2408.10001v6 (Wang & Mueller, 2026). Board-advancing code.

Evidence trail

RIS distance surrogate: d<=7 (both X and Z sides)
Witnesses confirmed by verifier's criteria

Tools

Source: arXiv:2408.10001v6
Verification: verify/validate_candidate.py

Parity checks

X-checks 30 (max weight 8) · Z-checks 30 (max weight 8)

H_X (30 checks, sparse supports)

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

H_Z (30 checks, sparse supports)

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

Code ID 60-12-7 · download JSON · raw on GitHub