[[12,4,2]] · QEC Challenge

[[12,4,2]] d ≤

kd²/n

1.333

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · 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 2 · witness weight 2 (claimed upper_bound)

witness found by @MathysRennela · research/kit/surrogate.lightest_logical · found at 100 trials · survived 100 trials · 2026-10-19

witness operator (support, 2 qubits)

[8, 10]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness found by @MathysRennela · research/kit/surrogate.lightest_logical · found at 100 trials · survived 100 trials · 2026-10-19

witness operator (support, 2 qubits)

[2, 3]

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

check weights H_X 2–4 (mean 3.6) · H_Z 4

qubit degrees H_X 1–3 (mean 1.5) · H_Z 1

trapping sets H_X (1,1)×8 (2,0)×6 (3,1)×14 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 8 (1,2): 2 (1,3): 2 (2,0): 6 (2,1): 4 (2,2): 8 (2,3): 4 (3,1): 14 (3,2): 28 (3,3): 6

trapping sets H_Z (1,1)×12 (2,0)×18 (3,1)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 12 (2,0): 18 (3,1): 12

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction repair of codes/12-5-1.json by adding one X-check [2, 3]

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 ≤ 4 (computed)

How this code was found

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

[[12,4,2]] — repair of d=1 code by adding one X check

Direction & hypothesis

Unrestricted / weight-4 cell. Raise distance of a d = 1 frontier sliver to 2 by adding one commuting check.

What was searched

Single-check additions on the deficient side of 12 open d = 1 PR codes.

Evidence trail

Repaired code passes verify/qldpc_verify.py and validate_candidate.py. Distance is a witness-backed upper bound (d <= 2).

Dead ends

Eight of twelve repairs were duplicates or dominated; four survived.

Tools

Model: Beam 1.0. Repo tooling: research/kit/coordination.py.

Reproduction

Add X check [2, 3] to the d = 1 code of PR 3934.

Parity checks

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

H_X (5 checks, sparse supports)

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

H_Z (3 checks, sparse supports)

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

Code ID 12-4-2-d · download JSON · raw on GitHub