[[25,1,5]] · QEC Challenge

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[[25,1,5]] d =

kd²/n

1.0

X/Z

1.4142

layers

swaps

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 5 · witness weight 5 (claimed exact)

witness operator (support, 5 qubits)

[2, 3, 5, 6, 9]

d_Z 5 · witness weight 5 (claimed exact)

witness operator (support, 5 qubits)

[0, 5, 10, 16, 21]

certificate exact, d = 5 · scipy/HiGHS MILP

X: min over 1 Z-logical cosets = 5, all proven; Z: min over 1 X-logical cosets = 5, all proven

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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 2–4 (mean 3.333) · H_Z 2–4 (mean 3.333)

qubit degrees H_X 1–2 (mean 1.6) · H_Z 1–2 (mean 1.6)

trapping sets H_X (1,1)×10 (2,0)×4 (3,1)×46 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 10 (1,2): 15 (2,0): 4 (2,1): 18 (2,2): 30 (3,1): 46 (3,2): 61 (3,3): 8 (3,4): 16

trapping sets H_Z (1,1)×10 (2,0)×4 (3,1)×46 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 10 (1,2): 15 (2,0): 4 (2,1): 18 (2,2): 30 (3,1): 46 (3,2): 61 (3,3): 8 (3,4): 16

witness diameter X 4.0 · Z 4.1231 (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)

Circuit tier

syndrome-extraction memory circuits committed under circuits/25-1-5/ · canonical noise recipe, 5 rounds, stim 1.16.0

d_circ ≤ 5 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 5 · fault-set witness of 5 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 5)

[42, 44, 106, 110, 157]

d_circ^Z 5 · fault-set witness of 5 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 5)

[1155, 1156, 1379, 1558, 1559]

basis	p	ler/round	95% CI	failures/shots	status
X	0.01	0.11	[0.105, 0.115]	3560/10⁴	verified at PR time
X	0.005	0.0231	[0.0217, 0.0246]	1054/10⁴	verified post-merge (1077/9999 replica shots, 2026-10-19)
X	0.002	0.00208	[0.00181, 0.00238]	206/2×10⁴	verified post-merge (131/1.5×10<sup>4</sup> replica shots, 2026-10-19)
X	0.001	0.0003	[0.000256, 0.000352]	150/10⁵	verified post-merge (158/10×10<sup>4</sup> replica shots, 2026-10-19)
Z	0.01	0.107	[0.102, 0.112]	3503/10⁴	verified at PR time
Z	0.005	0.024	[0.0226, 0.0256]	1092/10⁴	verified post-merge (1119/9999 replica shots, 2026-10-19)
Z	0.002	0.00212	[0.00175, 0.00257]	105/10⁴	verified post-merge (151/1.4×10<sup>4</sup> replica shots, 2026-10-19)
Z	0.001	0.000254	[0.000214, 0.000303]	127/10⁵	verified post-merge (156/1.2×10<sup>5</sup> replica shots, 2026-10-19)

decoder bposd-cs-10; rates are compared across entries only at equal p; a point whose per-shot failure fraction would exceed 1/2 is not admitted

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 (25)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 0 nearest-neighbor SWAPs per round in total, at most 0 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 Kitaev, A. Yu.

provenance literature baseline

construction Rotated surface code, distance 5.

model classical construction (no AI model)

reference arXiv:quant-ph/9707021

date 1997-07-09

notes Distance is exact by construction. 2D layout added 2026-07-23: qubit q at (q//5, q%5) on the 5x5 planar grid; single layer, measured interaction radius sqrt(2).

family topological (a tag, not a ranking)

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

weight class weight ≤ 4 (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 12 (max weight 4) · Z-checks 12 (max weight 4)

H_X (12 checks, sparse supports)

[0, 1, 5, 6] [2, 3, 7, 8] [4, 9] [5, 10] [6, 7, 11, 12] [8, 9, 13, 14] [10, 11, 15, 16] [12, 13, 17, 18] [14, 19] [15, 20] [16, 17, 21, 22] [18, 19, 23, 24]

H_Z (12 checks, sparse supports)

[0, 1] [2, 3] [1, 2, 6, 7] [3, 4, 8, 9] [5, 6, 10, 11] [7, 8, 12, 13] [11, 12, 16, 17] [13, 14, 18, 19] [15, 16, 20, 21] [17, 18, 22, 23] [21, 22] [23, 24]

Code ID 25-1-5 · download JSON · raw on GitHub