[[36,2,6]] · QEC Challenge

[[36,2,6]] d ≤

kd²/n

2.0

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 4, w_Z = 4 (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 @mathys · research/kit/surrogate.lightest_logical · found at 8000 trials · survived 8000 trials · 2026-11-11

witness operator (support, 6 qubits)

[5, 15, 19, 24, 25, 34]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 8000 trials · survived 8000 trials · 2026-11-11

witness operator (support, 6 qubits)

[4, 5, 7, 13, 14, 18]

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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 4 · H_Z 4

qubit degrees H_X 2 · H_Z 2

trapping sets H_X (1,2)×36 (2,2)×108 (3,2)×324 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 36 (2,2): 108 (3,2): 324 (3,4): 72

trapping sets H_Z (1,2)×36 (2,2)×108 (3,2)×324 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 36 (2,2): 108 (3,2): 324 (3,4): 72

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction quantum Tanner code

model Beam 1.0 (claimed, not verified)

date 2026-11-11

family quantum Tanner (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 4 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[36,2,6]] — quantum Tanner code over PSL(2,5)

Direction & hypothesis

Aimed at the weight-4 unrestricted CSS cell with a small quantum Tanner code. PSL(2,5) with a length-2 repetition seed code yields n=36, d=6 at weight 4.

What was searched

Random quantum Tanner codes over PSL(2,5) with RepetitionCode(2), seeds 0..11, bipartite and non-bipartite variants, screened at 300 trials per side and re-screened at 10,000 trials per side.

Evidence trail

Witness-backed upper bound d<=6 on both X and Z sides. Full gate refutation passed on a fresh seed.

Dead ends

Rep-3 instances over the same group landed at w=6 with d around 3-6 and did not advance their cell. Non-bipartite variants did not improve d.

Tools

qLDPC library (QTCode.random), challenge research kit, gf2_fast. Model: Beam 1.0.

Reproduction

Group PSL(2,5), seed code repetition length 2, random seed 9, bipartite False.

Parity checks

X-checks 18 (max weight 4) · Z-checks 18 (max weight 4)

H_X (18 checks, sparse supports)

[0, 1, 2, 3] [12, 15, 16, 17] [29, 31, 32, 33] [13, 18, 19, 20] [4, 7, 8, 9] [21, 24, 25, 26] [17, 21, 22, 23] [2, 4, 5, 6] [15, 25, 27, 28] [0, 8, 10, 11] [10, 12, 13, 14] [3, 27, 29, 30] [5, 16, 18, 34] [1, 22, 31, 35] [6, 19, 24, 30] [7, 14, 23, 32] [11, 20, 26, 35] [9, 28, 33, 34]

H_Z (18 checks, sparse supports)

[7, 9, 32, 33] [19, 20, 24, 26] [0, 2, 4, 8] [15, 17, 21, 25] [1, 3, 29, 31] [12, 13, 16, 18] [10, 11, 13, 20] [27, 28, 29, 33] [5, 6, 18, 19] [22, 23, 31, 32] [21, 22, 26, 35] [4, 5, 9, 34] [24, 25, 27, 30] [7, 8, 10, 14] [15, 16, 28, 34] [0, 1, 11, 35] [12, 14, 17, 23] [2, 3, 6, 30]

Code ID 36-2-6-b · download JSON · raw on GitHub