[[12,5,2]] · QEC Challenge

[[12,5,2]] d ≤

kd²/n

1.667

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[1, 3]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[8, 10]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 5 · H_Z 5

qubit degrees H_X 1–2 (mean 1.25) · H_Z 1–3 (mean 1.667)

trapping sets H_X (1,1)×9 (2,0)×15 (3,0)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 9 (1,2): 3 (2,0): 15 (2,1): 12 (3,0): 12 (3,1): 22 (3,2): 7

trapping sets H_Z (1,1)×7 (2,0)×9 (3,1)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 7 (1,2): 2 (1,3): 3 (2,0): 9 (2,1): 6 (2,2): 12 (2,3): 6 (3,1): 28 (3,2): 54 (3,3): 10

Circuit tier

syndrome-extraction memory circuits committed under circuits/12-5-2/ · canonical noise recipe, 2 rounds, stim 1.16.0

d_circ ≤ 2 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 2 · fault-set witness of 2 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 2)

[0, 2]

d_circ^Z 2 · fault-set witness of 2 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 2)

[11, 12]

no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction C-DELETE-CHECK a1 from codes/12-4-2-b.json (priced construction calculus)

model Beam 1.0 (claimed, not verified)

date 2026-10-19

family other (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

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

[[12,5,2]] Priced-construction deletion from 12-4-2-b

Direction & hypothesis

Apply a single C-DELETE-CHECK (remove check a1) to the board entry codes/12-4-2-b.json using the priced construction calculus. Deleting one independent X check raises k from 4 to 5 while n stays 12; the witness-backed distance falls to 2, so the candidate advances the weight-6 / unrestricted cell on k and check weight rather than on distance.

What was searched

One operation on one parent. No support sweep beyond validity of the single deletion and the inherited plus supplied logical witnesses.

Evidence trail

Confirmation ladder: the parent distance is 2 (witness-backed); the child keeps a weight-2 X logical and a weight-2 Z logical after deletion, so d <= 2 on both sides. The gate's refutation search found no lighter logical.

Dead ends

None recorded; the deletion was valid on the first try.

Tools

Model: Beam 1.0. The local verdict is verify/validate_candidate.py; the bundle was assembled by ./qldpc submit.

Reproduction

Reconstruct codes/12-4-2-b.json, delete its first X check (a1), and recompute n, k, and the witness-backed distance. The resulting checks are the submitted ones.

Parity checks

X-checks 3 (max weight 5) · Z-checks 4 (max weight 5)

H_X (3 checks, sparse supports)

[0, 1, 7, 9, 11] [2, 3, 6, 8, 10] [4, 5, 6, 8, 10]

H_Z (4 checks, sparse supports)

[0, 2, 4, 6, 7] [1, 3, 5, 6, 7] [0, 2, 4, 8, 9] [0, 2, 4, 10, 11]

Code ID 12-5-2 · download JSON · raw on GitHub