[[52,13,5]] · QEC Challenge

[[52,13,5]] d ≤

kd²/n

6.25

X/Z

1.2

Share this result

Distance

X/Z asymmetry 1.2 · d_X ≤ 6, d_Z ≤ 5 · 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 6 · witness weight 6 (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, 6 qubits)

[12, 18, 21, 30, 35, 39]

d_Z 5 · witness weight 5 (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, 5 qubits)

[3, 18, 24, 26, 51]

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 7.842) · H_Z 6–8 (mean 7.7)

qubit degrees H_X 1–5 (mean 2.865) · H_Z 1–5 (mean 2.962)

trapping sets H_X (1,1)×2 (2,1)×5 (3,1)×23 (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): 21 (1,3): 16 (1,4): 8 (1,5): 5 (2,1): 5 (2,2): 44 (2,3): 102 (2,4): 116 (2,5): 122 (2,6): 63 (2,7): 19 (2,8): 7 (3,1): 23 (3,2): 126 (3,3): 506 (3,4): 1005 (3,5): 1421 (3,6): 1387 (3,7): 910 (3,8): 518 (3,9): 234 (3,10): 67 (3,11): 7 (3,12): 3

trapping sets H_Z (1,1)×3 (2,1)×5 (3,1)×17 (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): 17 (1,3): 16 (1,4): 11 (1,5): 5 (2,1): 5 (2,2): 31 (2,3): 83 (2,4): 116 (2,5): 139 (2,6): 74 (2,7): 30 (2,8): 6 (3,1): 17 (3,2): 86 (3,3): 372 (3,4): 866 (3,5): 1350 (3,6): 1466 (3,7): 1176 (3,8): 665 (3,9): 294 (3,10): 93 (3,11): 19 (3,12): 2

Circuit tier

syndrome-extraction memory circuits committed under circuits/52-13-5/ · canonical noise recipe, 5 rounds, stim 1.16.0

d_circ ≤ 3 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 3 · fault-set witness of 3 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 3)

[9579, 9580, 9639]

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

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

[6848, 6853, 7385, 7407]

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

[[52,13,5]] — 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 53-13-5.

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 19 (max weight 8) · Z-checks 20 (max weight 8)

H_X (19 checks, sparse supports)

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

H_Z (20 checks, sparse supports)

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

Code ID 52-13-5 · download JSON · raw on GitHub