[[56,8,8]] · QEC Challenge

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[[56,8,8]] d =

kd²/n

9.143

X/Z

0.0152

7.0

layers

swaps

383

Share this result

Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[7, 9, 26, 32, 33, 36, 38, 41]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[0, 15, 19, 33, 37, 39, 47, 54]

certificate exact, d = 8 · scipy/HiGHS MILP

X: no logical < 8 exists; Z: no logical < 8 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)×56 (2,4)×140 (3,2)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 56 (2,4): 140 (2,6): 504 (3,2): 28 (3,4): 336 (3,6): 3052 (3,8): 5544 (3,10): 448

trapping sets H_Z (1,4)×56 (2,4)×140 (3,2)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 56 (2,4): 140 (2,6): 504 (3,2): 28 (3,4): 336 (3,6): 3052 (3,8): 5544 (3,10): 448

witness diameter X 7.6158 · Z 7.0711 (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 (56)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 383 nearest-neighbor SWAPs per round in total, at most 11 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_4xZ_7, bivariate A=[(0, 0), (1, 1), (2, 2), (0, 4)], B=[(0, 0), (2, 2), (2, 6), (3, 5)]

model MiMo V2.5 (claimed, not verified)

date 2026-09-15

family bivariate bicycle (a tag, not a ranking)

locality 2D-local bilayer (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

[[56,8,8]] — from arXiv:2408.10001v6

Direction & hypothesis

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

Evidence trail

RIS distance surrogate: d<=8 (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 28 (max weight 8) · Z-checks 28 (max weight 8)

H_X (28 checks, sparse supports)

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

H_Z (28 checks, sparse supports)

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

Code ID 56-8-8 · download JSON · raw on GitHub