[[60,18,6]] · QEC Challenge

[[60,18,6]] d ≤

kd²/n

10.8

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · 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)

[1, 11, 12, 19, 49, 58]

d_Z 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)

[15, 20, 24, 25, 40, 53]

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 7–8 (mean 7.81) · H_Z 7–8 (mean 7.714)

qubit degrees H_X 1–4 (mean 2.733) · H_Z 1–4 (mean 2.7)

trapping sets H_X (1,1)×2 (2,1)×4 (3,1)×16 (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): 22 (1,3): 26 (1,4): 10 (2,1): 4 (2,2): 38 (2,3): 148 (2,4): 218 (2,5): 124 (2,6): 28 (3,1): 16 (3,2): 127 (3,3): 659 (3,4): 1699 (3,5): 2228 (3,6): 1688 (3,7): 827 (3,8): 305 (3,9): 65 (3,10): 9

trapping sets H_Z (1,1)×3 (2,1)×3 (3,1)×21 (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): 23 (1,3): 23 (1,4): 11 (2,1): 3 (2,2): 45 (2,3): 146 (2,4): 189 (2,5): 135 (2,6): 28 (3,1): 21 (3,2): 152 (3,3): 651 (3,4): 1585 (3,5): 2120 (3,6): 1575 (3,7): 863 (3,8): 296 (3,9): 79 (3,10): 5

Circuit tier

syndrome-extraction memory circuits committed under circuits/60-18-6/ · canonical noise recipe, 6 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)

[12309, 12320, 12378, 12672]

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

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

[5433, 5511, 5771, 6234]

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

[[60,18,6]] — 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 61-18-6.

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, 8, 12, 29, 30, 39, 52] [0, 1, 18, 22, 31, 42, 45, 56] [0, 2, 17, 19, 32, 38, 48, 55] [3, 19, 22, 26, 33, 41, 50] [1, 4, 23, 25, 34, 40, 48, 51] [2, 5, 20, 24, 31, 35, 47, 59] [1, 2, 3, 6, 30, 36, 54, 58] [3, 7, 12, 25, 37, 47, 53, 55] [4, 8, 26, 28, 35, 38, 46, 54] [5, 6, 9, 10, 32, 34, 39] [3, 8, 10, 24, 40, 45, 49, 57] [4, 6, 7, 11, 31, 41, 44] [5, 12, 26, 27, 36, 42, 43, 51] [7, 13, 18, 28, 32, 43, 57, 58] [10, 14, 17, 27, 37, 44, 54, 56] [9, 11, 15, 16, 36, 38, 45, 53] [0, 7, 10, 16, 33, 46, 51, 59] [8, 11, 13, 17, 34, 42, 47, 50] [9, 12, 14, 18, 35, 41, 48, 49] [13, 19, 23, 29, 36, 44, 49, 59] [15, 17, 20, 21, 30, 41, 51, 57]

H_Z (21 checks, sparse supports)

[0, 6, 20, 28, 30, 31, 32, 46] [1, 5, 11, 29, 31, 34, 36, 52] [2, 9, 13, 26, 32, 35, 36, 50] [3, 16, 23, 27, 33, 36, 37, 40] [4, 9, 17, 22, 34, 38, 41, 56] [5, 8, 18, 21, 35, 39, 42, 57] [6, 12, 15, 19, 36, 39, 41, 55] [7, 14, 21, 26, 37, 41, 43, 46] [2, 8, 15, 25, 30, 38, 40, 47] [0, 9, 23, 24, 39, 45, 48, 59] [4, 10, 29, 39, 40, 44, 46] [3, 11, 18, 20, 41, 45, 47, 58] [1, 12, 17, 30, 37, 42, 48] [12, 13, 24, 43, 47, 49, 52] [11, 14, 19, 25, 44, 48, 50, 53] [8, 16, 22, 45, 46, 50, 52] [5, 7, 17, 23, 32, 44, 47, 51] [2, 4, 18, 27, 31, 43, 48, 54] [10, 18, 19, 32, 33, 49, 56] [3, 17, 28, 50, 54, 55, 57] [0, 21, 25, 27, 51, 52, 55, 56]

Code ID 60-18-6 · download JSON · raw on GitHub