[[50,6,4]] · QEC Challenge

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[[50,6,4]] d =

kd²/n

1.92

X/Z

0.0768

2.2361

layers

swaps

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · 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 4 · witness weight 4 (claimed exact)

witness operator (support, 4 qubits)

[0, 20, 31, 40]

d_Z 4 · witness weight 4 (claimed exact)

witness operator (support, 4 qubits)

[11, 13, 36, 38]

certificate exact, d = 4 · scipy/HiGHS cutoff IP

X: no logical < 4 exists; Z: no logical < 4 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 6 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 2–6 (mean 4.087) · H_Z 2–6 (mean 4.261)

qubit degrees H_X 1–4 (mean 1.88) · H_Z 1–5 (mean 1.96)

trapping sets H_X (1,1)×18 (2,0)×6 (3,1)×84 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 18 (1,2): 21 (1,3): 10 (1,4): 1 (2,0): 6 (2,1): 22 (2,2): 69 (2,3): 56 (2,4): 24 (2,5): 2 (3,1): 84 (3,2): 231 (3,3): 271 (3,4): 230 (3,5): 122 (3,6): 44 (3,7): 7

trapping sets H_Z (1,1)×19 (2,0)×8 (3,1)×93 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 19 (1,2): 17 (1,3): 12 (1,4): 1 (1,5): 1 (2,0): 8 (2,1): 26 (2,2): 54 (2,3): 59 (2,4): 22 (2,5): 8 (2,6): 5 (2,7): 1 (3,1): 93 (3,2): 205 (3,3): 225 (3,4): 237 (3,5): 136 (3,6): 79 (3,7): 33 (3,8): 11

witness diameter X 4.0 · Z 2.0 (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/50-6-4/ · canonical noise recipe, 4 rounds, stim 1.16.0

d_circ ≤ 4 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 4 · fault-set witness of 4 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 4)

[851, 966, 1046, 1600]

d_circ^Z 4 · fault-set witness of 4 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 4)

[1811, 1812, 3384, 3387]

basis	p	ler/round	95% CI	failures/shots	status
X	0.005	0.0906	[0.0867, 0.0947]	2752/10⁴	verified at PR time
X	0.002	0.00989	[0.00894, 0.0109]	384/10⁴	verified post-merge (373/9999 replica shots, 2026-10-19)
X	0.001	0.00186	[0.00166, 0.00209]	296/4×10⁴	verified post-merge (148/2×10<sup>4</sup> replica shots, 2026-10-19)
Z	0.005	0.116	[0.111, 0.121]	3261/10⁴	verified at PR time
Z	0.002	0.0129	[0.0118, 0.0141]	497/10⁴	verified post-merge (487/9999 replica shots, 2026-10-19)
Z	0.001	0.00265	[0.00241, 0.00292]	421/4×10⁴	verified post-merge (186/1.4×10<sup>4</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)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 54 nearest-neighbor SWAPs per round in total, at most 2 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, (5,5) 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 23 (max weight 6) · Z-checks 23 (max weight 6)

H_X (23 checks, sparse supports)

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

H_Z (23 checks, sparse supports)

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

Code ID 50-6-4 · download JSON · raw on GitHub