[[53,13,6]] · QEC Challenge

[[53,13,6]] d ≤

kd²/n

8.83019

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)

[8, 12, 28, 33, 51, 52]

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)

[7, 12, 40, 46, 48, 50]

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 7.85) · H_Z 7–8 (mean 7.75)

qubit degrees H_X 1–5 (mean 2.962) · H_Z 1–5 (mean 2.925)

trapping sets H_X (1,1)×4 (2,1)×2 (3,1)×15 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 4 (1,2): 14 (1,3): 19 (1,4): 12 (1,5): 4 (2,1): 2 (2,2): 33 (2,3): 90 (2,4): 129 (2,5): 140 (2,6): 79 (2,7): 26 (2,8): 4 (3,1): 15 (3,2): 97 (3,3): 385 (3,4): 934 (3,5): 1494 (3,6): 1601 (3,7): 1166 (3,8): 684 (3,9): 270 (3,10): 81 (3,11): 14 (3,12): 2

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): 18 (1,4): 11 (1,5): 4 (2,1): 5 (2,2): 32 (2,3): 93 (2,4): 128 (2,5): 138 (2,6): 68 (2,7): 24 (2,8): 3 (3,1): 17 (3,2): 97 (3,3): 417 (3,4): 969 (3,5): 1440 (3,6): 1508 (3,7): 1109 (3,8): 592 (3,9): 251 (3,10): 64 (3,11): 12

Circuit tier

syndrome-extraction memory circuits committed under circuits/53-13-6-b/ · canonical noise recipe, 6 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)

[12103, 12113, 12147]

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

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

[2764, 2765, 3525]

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

[[53,13,6]] — 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 54-13-6.

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

H_X (20 checks, sparse supports)

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

H_Z (20 checks, sparse supports)

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

Code ID 53-13-6-b · download JSON · raw on GitHub