[[180,94,4]] · QEC Challenge

[[180,94,4]] d ≤

180

kd²/n

8.356

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 4, d_Z ≤ 4 · 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 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[26, 30, 73, 132]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[62, 94, 137, 175]

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 12 · H_Z 12

qubit degrees H_X 3 · H_Z 3

trapping sets H_X (1,3)×180 (2,2)×210 (3,3)×5545 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 180 (2,2): 210 (2,4): 2550 (3,3): 5545 (3,5): 48185 (3,7): 6275

trapping sets H_Z (1,3)×180 (2,2)×210 (3,3)×5545 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 180 (2,2): 210 (2,4): 2550 (3,3): 5545 (3,5): 48185 (3,7): 6275

Circuit tier

syndrome-extraction memory circuits committed under circuits/180-94-4-b/ · canonical noise recipe, 4 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)

[15168, 15169, 20102]

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

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

[19885, 19890, 22974]

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 Cornucopia code q=5 (arXiv:2610.19073v2) with column shifts sA=[4,0,3,3,4,0], sB=[1,3,3,2,1,0] found by a 200-tuple random shift search.

model Beam 1.0 (claimed, not verified)

date 2026-10-21

notes Distinct code filed as codes/180-94-4-b.json: WL-signature match to 180-94-4.json but a different parity-check fingerprint (different column-shift tuple of the same Cornucopia construction), so both are separate entries on the Pareto frontier.

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

[[180,94,4]] — Cornucopia q=5 with searched column shifts

Direction & hypothesis

Target cell: weight-9plus × unrestricted on the CSS board (weight-12 checks, no layout), family other. The Cornucopia construction (Lu, Li, Ma et al., "A modular quantum computer based on Cornucopia codes", arXiv:2610.19073v2) fixes only the row maps; its twelve column shifts are free parameters. The hypothesis: at this admissible q (n = 36q ≤ 700) an unexplored shift tuple would produce a code that is not dominated by the Cornucopia instance of the same (n, k) already on the board.

What was searched

200-tuple random column-shift tuples at q=5 (sA, sB ∈ Z_5^6 each), every code screened with the gf2_fast randomized logical search at 500 trials, seeds from the sweep's own RNG stream. The best tuple, sA=[4, 0, 3, 3, 4, 0], sB=[1, 3, 3, 2, 1, 0], was then confirmed at 10,000 accelerator trials plus a 1,000-trial-per-side RIS pass.

Evidence trail

Ladder for the submitted code: 500-trial screen → d<=4; 10,000-trial accelerator pass → d<=4 (witnesses embedded in the entry); the gate's own refutation on a fresh seed found nothing lighter. The claim is a witness-backed upper bound, not an exact distance.

Dead ends

The same sweep at q=13 produced nothing above d ≤ 6 while the board already holds q=13 at d=8, so q=13 was dropped. Within this sweep, tuples screened at 500 trials routinely collapsed to d ≤ 4 at smaller q; none of the confirmed readings moved below the submitted bound.

Tools

Model: Beam 1.0. Repo tooling: research/kit/submit.make_submission, research/kit/surrogate (NumPy + gf2_fast backend), gate research/kit/coordination.py gate. Build matrices per arXiv:2610.19073v2 Methods, Eq. (1).

Reproduction

q=5, row maps ρ_A(0)=r+1, ρ_A(1)=2r+2, ρ_B(2)=r+1, ρ_B(3)=2r+2 (identity otherwise), column shifts sA=[4, 0, 3, 3, 4, 0], sB=[1, 3, 3, 2, 1, 0] on Z_5. Index data qubit (b,r,c) as (3b+r)q+c; HX block (i,j) = [A_(j-i) | B_(j-i)], HZ block (i,j) = [B_(i-j)^-1 | A_(i-j)^-1] for i∈{0,1,2}, j∈{0..5}.

Parity checks

X-checks 45 (max weight 12) · Z-checks 45 (max weight 12)

H_X (45 checks, sparse supports)

[9, 25, 33, 48, 64, 75, 91, 108, 128, 147, 151, 165] [5, 26, 34, 49, 60, 76, 92, 109, 129, 148, 152, 166] [6, 27, 30, 45, 61, 77, 93, 105, 125, 149, 153, 167] [7, 28, 31, 46, 62, 78, 94, 106, 126, 145, 154, 168] [8, 29, 32, 47, 63, 79, 90, 107, 127, 146, 150, 169] [14, 20, 38, 53, 69, 80, 96, 113, 133, 142, 156, 170] [10, 21, 39, 54, 65, 81, 97, 114, 134, 143, 157, 171] [11, 22, 35, 50, 66, 82, 98, 110, 130, 144, 158, 172] [12, 23, 36, 51, 67, 83, 99, 111, 131, 140, 159, 173] [13, 24, 37, 52, 68, 84, 95, 112, 132, 141, 155, 174] [4, 15, 43, 58, 74, 85, 101, 118, 123, 137, 161, 175] [0, 16, 44, 59, 70, 86, 102, 119, 124, 138, 162, 176] [1, 17, 40, 55, 71, 87, 103, 115, 120, 139, 163, 177] [2, 18, 41, 56, 72, 88, 104, 116, 121, 135, 164, 178] [3, 19, 42, 57, 73, 89, 100, 117, 122, 136, 160, 179] [0, 24, 40, 48, 63, 79, 90, 106, 123, 143, 162, 166] [1, 20, 41, 49, 64, 75, 91, 107, 124, 144, 163, 167] [2, 21, 42, 45, 60, 76, 92, 108, 120, 140, 164, 168] [3, 22, 43, 46, 61, 77, 93, 109, 121, 141, 160, 169] [4, 23, 44, 47, 62, 78, 94, 105, 122, 142, 161, 165] [5, 29, 35, 53, 68, 84, 95, 111, 128, 148, 157, 171] [6, 25, 36, 54, 69, 80, 96, 112, 129, 149, 158, 172] [7, 26, 37, 50, 65, 81, 97, 113, 125, 145, 159, 173] [8, 27, 38, 51, 66, 82, 98, 114, 126, 146, 155, 174] [9, 28, 39, 52, 67, 83, 99, 110, 127, 147, 156, 170] [10, 19, 30, 58, 73, 89, 100, 116, 133, 138, 152, 176] [11, 15, 31, 59, 74, 85, 101, 117, 134, 139, 153, 177] [12, 16, 32, 55, 70, 86, 102, 118, 130, 135, 154, 178] [13, 17, 33, 56, 71, 87, 103, 119, 131, 136, 150, 179] [14, 18, 34, 57, 72, 88, 104, 115, 132, 137, 151, 175] [4, 15, 39, 55, 63, 78, 91, 105, 121, 138, 158, 177] [0, 16, 35, 56, 64, 79, 92, 106, 122, 139, 159, 178] [1, 17, 36, 57, 60, 75, 93, 107, 123, 135, 155, 179] [2, 18, 37, 58, 61, 76, 94, 108, 124, 136, 156, 175] [3, 19, 38, 59, 62, 77, 90, 109, 120, 137, 157, 176] [9, 20, 44, 50, 68, 83, 96, 110, 126, 143, 163, 172] [5, 21, 40, 51, 69, 84, 97, 111, 127, 144, 164, 173] [6, 22, 41, 52, 65, 80, 98, 112, 128, 140, 160, 174] [7, 23, 42, 53, 66, 81, 99, 113, 129, 141, 161, 170] [8, 24, 43, 54, 67, 82, 95, 114, 125, 142, 162, 171] [14, 25, 34, 45, 73, 88, 101, 115, 131, 148, 153, 167] [10, 26, 30, 46, 74, 89, 102, 116, 132, 149, 154, 168] [11, 27, 31, 47, 70, 85, 103, 117, 133, 145, 150, 169] [12, 28, 32, 48, 71, 86, 104, 118, 134, 146, 151, 165] [13, 29, 33, 49, 72, 87, 100, 119, 130, 147, 152, 166]

H_Z (45 checks, sparse supports)

[4, 15, 34, 58, 72, 77, 101, 105, 121, 137, 152, 175] [0, 16, 30, 59, 73, 78, 102, 106, 122, 138, 153, 176] [1, 17, 31, 55, 74, 79, 103, 107, 123, 139, 154, 177] [2, 18, 32, 56, 70, 75, 104, 108, 124, 135, 150, 178] [3, 19, 33, 57, 71, 76, 100, 109, 120, 136, 151, 179] [9, 20, 39, 53, 62, 82, 91, 110, 126, 142, 157, 170] [5, 21, 35, 54, 63, 83, 92, 111, 127, 143, 158, 171] [6, 22, 36, 50, 64, 84, 93, 112, 128, 144, 159, 172] [7, 23, 37, 51, 60, 80, 94, 113, 129, 140, 155, 173] [8, 24, 38, 52, 61, 81, 90, 114, 125, 141, 156, 174] [14, 25, 44, 48, 67, 87, 96, 115, 131, 147, 162, 165] [10, 26, 40, 49, 68, 88, 97, 116, 132, 148, 163, 166] [11, 27, 41, 45, 69, 89, 98, 117, 133, 149, 164, 167] [12, 28, 42, 46, 65, 85, 99, 118, 134, 145, 160, 168] [13, 29, 43, 47, 66, 86, 95, 119, 130, 146, 161, 169] [2, 19, 30, 49, 73, 87, 100, 116, 120, 136, 152, 167] [3, 15, 31, 45, 74, 88, 101, 117, 121, 137, 153, 168] [4, 16, 32, 46, 70, 89, 102, 118, 122, 138, 154, 169] [0, 17, 33, 47, 71, 85, 103, 119, 123, 139, 150, 165] [1, 18, 34, 48, 72, 86, 104, 115, 124, 135, 151, 166] [7, 24, 35, 54, 68, 77, 95, 106, 125, 141, 157, 172] [8, 20, 36, 50, 69, 78, 96, 107, 126, 142, 158, 173] [9, 21, 37, 51, 65, 79, 97, 108, 127, 143, 159, 174] [5, 22, 38, 52, 66, 75, 98, 109, 128, 144, 155, 170] [6, 23, 39, 53, 67, 76, 99, 105, 129, 140, 156, 171] [12, 29, 40, 59, 63, 82, 90, 111, 130, 146, 162, 177] [13, 25, 41, 55, 64, 83, 91, 112, 131, 147, 163, 178] [14, 26, 42, 56, 60, 84, 92, 113, 132, 148, 164, 179] [10, 27, 43, 57, 61, 80, 93, 114, 133, 149, 160, 175] [11, 28, 44, 58, 62, 81, 94, 110, 134, 145, 161, 176] [12, 17, 34, 45, 64, 88, 92, 115, 131, 135, 151, 167] [13, 18, 30, 46, 60, 89, 93, 116, 132, 136, 152, 168] [14, 19, 31, 47, 61, 85, 94, 117, 133, 137, 153, 169] [10, 15, 32, 48, 62, 86, 90, 118, 134, 138, 154, 165] [11, 16, 33, 49, 63, 87, 91, 119, 130, 139, 150, 166] [2, 22, 39, 50, 69, 83, 97, 110, 121, 140, 156, 172] [3, 23, 35, 51, 65, 84, 98, 111, 122, 141, 157, 173] [4, 24, 36, 52, 66, 80, 99, 112, 123, 142, 158, 174] [0, 20, 37, 53, 67, 81, 95, 113, 124, 143, 159, 170] [1, 21, 38, 54, 68, 82, 96, 114, 120, 144, 155, 171] [7, 27, 44, 55, 74, 78, 102, 105, 126, 145, 161, 177] [8, 28, 40, 56, 70, 79, 103, 106, 127, 146, 162, 178] [9, 29, 41, 57, 71, 75, 104, 107, 128, 147, 163, 179] [5, 25, 42, 58, 72, 76, 100, 108, 129, 148, 164, 175] [6, 26, 43, 59, 73, 77, 101, 109, 125, 149, 160, 176]

Code ID 180-94-4-b · download JSON · raw on GitHub