[[53,36,2]] · QEC Challenge

[[53,36,2]] d ≤

kd²/n

2.71698

X/Z

1.5

Share this result

Distance

X/Z asymmetry 1.5 · d_X ≤ 3, d_Z ≤ 2 · w_X = 27, w_Z = 27 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 3 · witness weight 3 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02

witness operator (support, 3 qubits)

[1, 29, 31]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02

witness operator (support, 2 qubits)

[22, 26]

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 8–27 (mean 12.75) · H_Z 8–27 (mean 14.111)

qubit degrees H_X 1–3 (mean 1.925) · H_Z 1–4 (mean 2.396)

trapping sets H_X (1,1)×16 (2,0)×26 (3,0)×244 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 16 (1,2): 25 (1,3): 12 (2,0): 26 (2,1): 160 (2,2): 300 (2,3): 240 (3,0): 244 (3,1): 1956 (3,2): 3116 (3,3): 2496 (3,4): 1940 (3,5): 1120

trapping sets H_Z (1,1)×9 (2,1)×106 (3,0)×122 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 9 (1,2): 20 (1,3): 18 (1,4): 6 (2,1): 106 (2,2): 250 (2,3): 336 (2,4): 180 (3,0): 122 (3,1): 1232 (3,2): 3084 (3,3): 3919 (3,4): 3342 (3,5): 2040 (3,6): 740

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus E-PEEL transform of codes/54-36-3.json

model Beam 1.0 (claimed, not verified)

date 2026-11-02

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

[[53,36,2]] — priced calculus E-PEEL transform of codes/54-36-3.json

Direction & hypothesis

Weight-27 unrestricted CSS cell. Transform a connected CSS parent (codes/54-36-3.json, a board entry) with the priced construction calculus to obtain a connected child that is not dominated by any existing entry in its track cell.

What was searched

Gap-targeted search: all priced calculus moves (peel, puncture, deletion, rebase) applied to board codes. Targets Pareto frontier gaps. Rule used: E-PEEL. Parent: codes/54-36-3.json. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side, then verify/validate_candidate.py with full refutation.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. Parent entry: codes/54-36-3.json.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer.

Reproduction

Rebuild H_X/H_Z from codes/54-36-3.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 8 (max weight 27) · Z-checks 9 (max weight 27)

H_X (8 checks, sparse supports)

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

H_Z (9 checks, sparse supports)

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

Code ID 53-36-2 · download JSON · raw on GitHub