[[84,61,2]] · QEC Challenge

[[84,61,2]] d ≤

kd²/n

2.90476

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 2, d_Z ≤ 2 · w_X = 12, w_Z = 12 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[7, 54]

d_Z 2 · witness weight 2 (claimed upper_bound)

witness operator (support, 2 qubits)

[33, 41]

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 8–12 (mean 10.583) · H_Z 9–12 (mean 11.0)

qubit degrees H_X 1–3 (mean 1.512) · H_Z 1–3 (mean 1.44)

trapping sets H_X (1,1)×45 (2,0)×97 (3,0)×480 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 45 (1,2): 35 (1,3): 4 (2,0): 97 (2,1): 258 (2,2): 186 (2,3): 42 (2,4): 2 (3,0): 480 (3,1): 1459 (3,2): 1619 (3,3): 966 (3,4): 297 (3,5): 48 (3,6): 4

trapping sets H_Z (1,1)×52 (2,0)×117 (3,0)×578 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 52 (1,2): 27 (1,3): 5 (2,0): 117 (2,1): 247 (2,2): 164 (2,3): 57 (2,4): 4 (3,0): 578 (3,1): 1413 (3,2): 1721 (3,3): 1068 (3,4): 411 (3,5): 86 (3,6): 8

Circuit tier

syndrome-extraction memory circuits committed under circuits/84-61-2/ · canonical noise recipe, 2 rounds, stim 1.16.0

d_circ ≤ 2 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 2 · fault-set witness of 2 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 2)

[43, 44]

d_circ^Z 2 · fault-set witness of 2 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 2)

[161, 163]

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 Graft reduction (degree-1 qubit removal) of a 107-qubit CSS board entry, preserving k and d while reducing n.

model Beam 1.0 (claimed, not verified)

date 2026-11-04

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

[[84,61,2]] — graft reduction of a CSS code to 84 qubits

Direction & hypothesis

Targeted the weight-9plus x unrestricted cell: reduce the qubit count of an existing CSS entry while preserving k and d, extending the Pareto frontier on the n axis.

What was searched

One candidate produced by grafting degree-1 qubits (qubits contained in exactly one X or Z check, removed together with that check) from a 107-qubit CSS board entry. The graft preserved k = 61 and the distance bound d <= 2 while reducing n from 107 to 84. Check weight remains 12.

Evidence trail

Distance confirmation: the graft checks preserved the bound d <= 2 at every removal (surrogate, 200 trials per side). The submitted witnesses reproduce the bound: X: d <= 2 (upper_bound), Z: d <= 2 (upper_bound).

Dead ends

None recorded beyond standard screening rejections.

Tools

Model: Beam 1.0. Harness: local code calculus graft (degree-1 qubit removal). Repo tooling: verify/validate_candidate.py, verify/qldpc_verify.py. Compute: local CPU.

Reproduction

The construction is not reproducible from a committed script. The check matrices are given in codes/84-61-2.json; re-run the trusted gate with uv run python verify/validate_candidate.py codes/84-61-2.json.

Parity checks

X-checks 12 (max weight 12) · Z-checks 11 (max weight 12)

H_X (12 checks, sparse supports)

[3, 12, 13, 27, 34, 40, 48, 58, 67, 70] [5, 10, 16, 22, 37, 42, 51, 60, 64, 73, 79] [4, 9, 17, 23, 29, 35, 49, 59, 65, 74, 80] [1, 7, 30, 38, 46, 55, 63, 75, 83] [2, 8, 18, 25, 31, 39, 44, 53, 56, 61, 76, 81] [0, 6, 19, 26, 32, 45, 52, 54, 62, 77, 82] [11, 16, 22, 29, 35, 43, 51, 57, 68, 74, 79] [2, 6, 15, 25, 28, 33, 41, 47, 55, 61, 72, 82] [10, 19, 24, 35, 49, 66, 77, 79] [3, 9, 22, 29, 36, 43, 50, 57, 64, 75, 80] [4, 11, 13, 20, 30, 39, 44, 53, 60, 68, 70, 78] [12, 14, 21, 31, 45, 52, 59, 69, 71]

H_Z (11 checks, sparse supports)

[1, 7, 26, 32, 34, 46, 48, 54, 62, 83] [0, 6, 14, 25, 31, 33, 45, 47, 56, 61, 71, 82] [3, 10, 16, 24, 35, 40, 49, 57, 66, 73, 79] [9, 17, 22, 27, 36, 50, 58, 64, 74, 80] [5, 11, 18, 25, 27, 37, 41, 51, 60, 67, 76, 81] [4, 12, 19, 26, 28, 35, 40, 49, 59, 68, 77, 82] [5, 8, 13, 20, 38, 40, 53, 60, 63] [4, 6, 14, 21, 27, 39, 52, 59, 61, 70, 78] [2, 10, 16, 23, 36, 42, 51, 56, 66, 72, 80] [1, 15, 22, 29, 35, 43, 50, 55, 65, 74, 79] [3, 12, 18, 26, 30, 33, 44, 48, 58, 69, 75, 82]

Code ID 84-61-2 · download JSON · raw on GitHub