[[54,12,7]] · QEC Challenge

[[54,12,7]] d ≤

kd²/n

10.8889

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · 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 7 · witness weight 7 (claimed upper_bound)

witness found by @MathysRennela · research/kit/surrogate.lightest_logical · found at 300 trials · survived 300 trials · 2026-11-07

witness operator (support, 7 qubits)

[1, 11, 27, 30, 35, 36, 38]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness found by @MathysRennela · research/kit/surrogate.lightest_logical · found at 300 trials · survived 300 trials · 2026-11-07

witness operator (support, 7 qubits)

[3, 4, 11, 25, 34, 40, 45]

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 7–8 (mean 7.857) · H_Z 7–8 (mean 7.857)

qubit degrees H_X 1–5 (mean 3.056) · H_Z 1–5 (mean 3.056)

trapping sets H_X (1,1)×2 (2,1)×2 (3,1)×11 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 2 (1,2): 18 (1,3): 19 (1,4): 5 (1,5): 10 (2,1): 2 (2,2): 30 (2,3): 94 (2,4): 106 (2,5): 123 (2,6): 120 (2,7): 24 (2,8): 27 (3,1): 11 (3,2): 81 (3,3): 363 (3,4): 840 (3,5): 1271 (3,6): 1700 (3,7): 1363 (3,8): 888 (3,9): 572 (3,10): 153 (3,11): 74 (3,12): 5 (3,13): 4

trapping sets H_Z (1,1)×3 (2,1)×3 (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): 3 (1,2): 16 (1,3): 16 (1,4): 13 (1,5): 6 (2,1): 3 (2,2): 30 (2,3): 74 (2,4): 120 (2,5): 159 (2,6): 90 (2,7): 44 (2,8): 8 (3,1): 12 (3,2): 73 (3,3): 337 (3,4): 789 (3,5): 1450 (3,6): 1709 (3,7): 1418 (3,8): 941 (3,9): 436 (3,10): 146 (3,11): 26 (3,12): 6

Circuit tier

syndrome-extraction memory circuits committed under circuits/54-12-7-b/ · canonical noise recipe, 7 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)

[13334, 13338, 13790, 14030]

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

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

[3036, 3044, 3263, 3482]

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 code-calculus v0.1 E-PEEL of a CSS parent

model Beam 1.0 (claimed, not verified)

date 2026-11-07

family other (a tag, not a ranking)

locality unrestricted (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

[[54,12,7]] — code-calculus v0.1 E-PEEL of a CSS parent

Direction & hypothesis

Targeted a Pareto opening in the unrestricted / weight-8 cell via the priced construction calculus (unitaryfoundation/code-calculus v0.1): E-PEEL a qubit (or check) of a CSS parent, recomputing ranks, connectivity, and max check weight. The child is board-advancing when its (n,k,d,w) is nondominated in its cell.

What was searched

Single-step E-PEEL moves over CSS parents, screened structurally (no bare qubit, connected Tanner graph, independent displayed rows) and by an optimistic 4-axis domination test against the board before any witness search. Survivors were packaged with research/kit/submit.py (make_submission, save_submission) and validated with research/kit/coordination.py gate (passed: true).

Evidence trail

Distance is a witness-backed upper bound from a 300-trial RIS screen per side (confidence upper_bound); no lower bound is claimed. Move: E-PEEL of parent 55-12-7.

Dead ends

Other moves and parents in the same sweep were dominated in their cell and discarded before gating.

Tools

Model: Beam 1.0. Author: @mathysrennela. Harness: OpenCode. Repo tooling: research/kit/submit.py, research/kit/coordination.py, research/kit/prescreen.py, verify/validate_candidate.py.

Reproduction

Load the parent CSS entry, build a CodeState via CodeState.from_challenge, enumerate E-PEEL candidates with the code-calculus planner, apply with apply_operation, and package the surviving child with make_submission. Library: github.com/MathysRennela/code-calculus (v0.1).

Parity checks

X-checks 21 (max weight 8) · Z-checks 21 (max weight 8)

H_X (21 checks, sparse supports)

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

H_Z (21 checks, sparse supports)

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

Code ID 54-12-7-b · download JSON · raw on GitHub