[[62,21,6]] · QEC Challenge

[[62,21,6]] d ≤

kd²/n

12.1935

X/Z

1.17

Share this result

Distance

X/Z asymmetry 1.17 · d_X ≤ 6, d_Z ≤ 7 · w_X = 16, w_Z = 16 (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 @vprusso · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-11-03

witness operator (support, 6 qubits)

[19, 26, 27, 48, 55, 60]

d_Z 7 · witness weight 7 (claimed upper_bound)

witness found by @vprusso · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-11-03

witness operator (support, 7 qubits)

[0, 8, 9, 18, 29, 39, 52]

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 14–16 (mean 15.143) · H_Z 16

qubit degrees H_X 2–9 (mean 5.129) · H_Z 2–8 (mean 5.161)

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

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

(1,2): 2 (1,3): 9 (1,4): 5 (1,5): 21 (1,6): 16 (1,7): 7 (1,8): 1 (1,9): 1 (2,1): 1 (2,2): 4 (2,3): 15 (2,4): 58 (2,5): 112 (2,6): 232 (2,7): 300 (2,8): 290 (2,9): 214 (2,10): 121 (2,11): 43 (2,12): 12 (2,13): 2 (2,14): 2 (3,1): 1 (3,2): 13 (3,3): 102 (3,4): 320 (3,5): 968 (3,6): 2233 (3,7): 3751 (3,8): 5233 (3,9): 5756 (3,10): 5104 (3,11): 3725 (3,12): 2133 (3,13): 943 (3,14): 342 (3,15): 92 (3,16): 17 (3,17): 2

trapping sets H_Z (1,2)×1 (2,2)×5 (3,1)×5 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 1 (1,3): 7 (1,4): 11 (1,5): 19 (1,6): 12 (1,7): 10 (1,8): 2 (2,2): 5 (2,3): 23 (2,4): 55 (2,5): 125 (2,6): 261 (2,7): 316 (2,8): 273 (2,9): 208 (2,10): 123 (2,11): 47 (2,12): 10 (3,1): 5 (3,2): 11 (3,3): 105 (3,4): 455 (3,5): 1183 (3,6): 2566 (3,7): 4239 (3,8): 5671 (3,9): 5962 (3,10): 5132 (3,11): 3446 (3,12): 1778 (3,13): 759 (3,14): 230 (3,15): 41 (3,16): 7

Circuit tier

syndrome-extraction memory circuits committed under circuits/62-21-6/ · canonical noise recipe, 6 rounds, stim 1.16.0

d_circ ≤ 4 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 5 · fault-set witness of 5 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 5)

[4399, 4427, 4455, 4499, 11256]

d_circ^Z 4 · fault-set witness of 4 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 4)

[8045, 8067, 8105, 8136]

no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Construction & provenance

authors @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction Priced construction calculus C-DELETE-CHECK of codes/62-20-6.json ([[62,20,6]], max check weight 16; itself Mathys Rennela's priced-calculus E-PEEL of codes/64-20-8.json, the depth-one universal code grown from the [[24,8,4]] seed in arXiv:2610.06730): delete X check a1 (rows numbered in file order after dropping dependent rows, as machine/priced_calculus.py labels them) and keep every qubit at its parent index. Consumer: github.com/unitaryfoundation/code-calculus @ 1b9a942, machine/priced_calculus.py.

model Claude Fable 5.1 (claimed, not verified)

builds on arXiv:2610.06730

date 2026-11-04

notes Single-step priced-calculus C-DELETE-CHECK child of board entry 62-20-6. verify/validate_candidate.py found no board entry sharing this code's fingerprint or Weisfeiler-Leman signature, local-Hadamard images included.

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

[[62,21,6]] priced calculus C-DELETE-CHECK of codes/62-20-6.json

Direction & hypothesis

Unrestricted weight-9plus CSS cell. The priced construction calculus was applied one move deep to every CSS entry on the board, and this code is the C-DELETE-CHECK of codes/62-20-6.json ([[62,20,6]], max check weight 16; itself Mathys Rennela's priced-calculus E-PEEL of codes/64-20-8.json, the depth-one universal code grown from the [[24,8,4]] seed in arXiv:2610.06730). Deleting X check a1 raises k from 20 to 21 and keeps the witnessed distance bound at 6; n = 62 and the max check weight 16 are unchanged.

What was searched

Every C-DELETE-CHECK, E-PEEL, and E-PUNCTURE move on each of the 2,783 CSS board entries as of 2026-11-02, 1,684,787 moves in all, with the consumer at github.com/unitaryfoundation/code-calculus @ 1b9a942. Children with a bare qubit, a disconnected Tanner graph, a Weisfeiler-Leman signature equal to a board entry or to an earlier child, or parameters already dominated inside the parent's cell were dropped before any search, leaving 334,191 distinct children. Children with n <= 60 and an inherited distance bound of at least 2, and children with 61 <= n <= 100 and a bound of at least 5, were screened with research/kit/surrogate.lightest_logical at 2,000 RIS trials per side; the survivors the kit's novelty pre-check called board-advancing were staged with research/kit/submit.py and gated with verify/validate_candidate.py, one representative per distinct (n, k, d, max weight, locality) point. This code is that representative for [[62,21,6]] at max check weight 16. The thirteen points audited first went to the board on 2026-11-04 (PRs #4588 to #4600); this one is from the remainder, re-checked against the board of 2026-11-04 with the gate's structural, deduplication, and novelty stages before the ladder below was run. The screen distance is a witness-backed upper bound; the rungs below are the independent confirmation runs on the staged file.

Evidence trail

Confirmation ladder for the submitted code, each rung a fresh seed unless marked otherwise, reading the lightest logical found:

| budget (trials/side) | lightest logical | | ---: | ---: | | 2,000 (staging screen, seeds 1 and 2) | 6 | | 1,000,000 (seed 101) | 6 | | 1,000,000 (seed 102) | 6 | | 10,000,000 (seed 103) | 6 |

4 fresh-seed rung(s) agree at the best bound, the deepest of them at 10,000,000 trials per side. The submitted witness reproduces it: X: d <= 6 (upper_bound), Z: d <= 7 (upper_bound).

Dead ends

None recorded for this candidate.

Tools

Model: Claude Fable 5.1 (Claude Code). Repo tooling: research/kit/coordination.py, research/kit/submit.py, research/kit/surrogate.py, research/audits/leader_audit.py. The local verdict is verify/validate_candidate.py; the bundle was assembled by research/kit/promote.py.

Reproduction

Load codes/62-20-6.json and delete X check a1, where the X rows are numbered a1, a2, ... in file order after dropping rows dependent on earlier ones, as CodeState.from_challenge in machine/priced_calculus.py labels them; every qubit keeps its index. In github.com/unitaryfoundation/code-calculus @ 1b9a942 this is apply_operation(state, {'op': 'C-DELETE-CHECK', 'row_id': 'a1'}). Then run research/audits/leader_audit.py ladder on the result with --ladder 1000000:101,102 10000000:103 to reproduce the confirmation rungs; the witnesses in the entry are the ones those runs validated.

Parity checks

X-checks 21 (max weight 16) · Z-checks 20 (max weight 16)

H_X (21 checks, sparse supports)

[5, 7, 12, 14, 16, 18, 24, 26, 31, 33, 40, 42, 52, 53, 58, 60] [32, 34, 35, 37, 40, 42, 43, 45, 48, 50, 51, 55, 57, 58, 60] [2, 4, 9, 15, 17, 22, 25, 27, 34, 36, 39, 45, 47, 57, 59] [0, 1, 8, 9, 18, 25, 26, 35, 36, 43, 44, 53, 60, 61] [3, 5, 10, 12, 20, 25, 27, 31, 37, 40, 46, 47, 55, 61] [1, 7, 10, 12, 20, 23, 29, 34, 36, 39, 45, 48, 53, 56, 58] [4, 6, 12, 14, 17, 24, 26, 32, 34, 40, 42, 51, 58, 60] [3, 7, 11, 15, 18, 21, 25, 29, 31, 35, 39, 43, 48, 52, 55, 59] [0, 3, 12, 15, 16, 27, 30, 35, 38, 39, 42, 51, 53, 54, 57] [17, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46] [3, 4, 9, 14, 17, 21, 26, 27, 31, 38, 41, 44, 49, 52, 54, 61] [2, 7, 11, 14, 18, 22, 26, 29, 34, 37, 41, 46, 50, 56, 61] [2, 5, 8, 15, 19, 24, 29, 31, 38, 41, 44, 48, 57, 58] [4, 6, 8, 10, 17, 28, 30, 31, 33, 43, 45, 52, 53, 55, 57] [0, 2, 13, 15, 19, 21, 24, 26, 36, 38, 39, 41, 48, 50, 58, 60] [9, 10, 12, 15, 17, 18, 19, 22, 39, 42, 44, 45, 47, 50, 52] [8, 9, 10, 11, 19, 20, 21, 22, 43, 44, 45, 46, 47, 48, 49, 50] [1, 2, 9, 10, 16, 23, 26, 35, 38, 43, 46, 52, 59, 60] [1, 3, 5, 7, 8, 10, 12, 14, 16, 18, 19, 21, 24, 26, 28, 30] [1, 3, 9, 11, 16, 18, 23, 25, 32, 34, 40, 42, 47, 49, 54, 56] [4, 6, 13, 15, 20, 22, 27, 29, 32, 34, 39, 41, 47, 49, 55, 57]

H_Z (20 checks, sparse supports)

[2, 3, 4, 5, 16, 17, 21, 22, 39, 40, 45, 46, 56, 57, 58, 59] [5, 7, 9, 11, 20, 22, 24, 26, 32, 34, 44, 46, 48, 50, 59, 61] [0, 2, 5, 7, 8, 10, 13, 15, 47, 49, 52, 53, 54, 56, 59, 61] [4, 5, 6, 7, 19, 20, 21, 22, 39, 40, 41, 42, 54, 55, 56, 57] [9, 10, 13, 14, 17, 18, 20, 21, 32, 33, 36, 37, 55, 56, 59, 60] [8, 9, 14, 15, 25, 26, 27, 28, 31, 32, 37, 38, 49, 50, 51, 52] [1, 3, 13, 15, 16, 18, 27, 29, 31, 33, 43, 45, 48, 50, 59, 61] [2, 6, 10, 14, 16, 19, 23, 27, 33, 37, 41, 45, 47, 51, 54, 58] [2, 4, 8, 14, 18, 19, 23, 29, 33, 35, 39, 45, 49, 51, 54, 60] [1, 4, 11, 14, 16, 20, 25, 30, 34, 37, 40, 43, 49, 53, 54, 59] [0, 3, 5, 6, 24, 25, 27, 30, 32, 33, 35, 38, 54, 57, 59, 60] [2, 7, 8, 13, 16, 20, 25, 30, 33, 38, 39, 44, 47, 52, 56, 61] [1, 7, 9, 15, 17, 22, 24, 30, 33, 35, 41, 43, 49, 51, 56, 58] [10, 11, 12, 13, 23, 24, 29, 30, 41, 42, 43, 44, 54, 55, 60, 61] [2, 3, 12, 13, 21, 22, 23, 24, 31, 32, 45, 46, 51, 52, 56, 57] [0, 5, 11, 14, 16, 20, 26, 29, 33, 38, 40, 43, 49, 53, 55, 58] [4, 7, 8, 11, 17, 18, 28, 29, 36, 37, 40, 41, 47, 50, 58, 61] [0, 6, 8, 14, 18, 19, 25, 27, 34, 36, 42, 44, 48, 53, 55, 61] [2, 3, 12, 13, 16, 17, 29, 30, 31, 32, 45, 46, 49, 50, 58, 59] [5, 7, 9, 11, 19, 21, 23, 25, 35, 37, 39, 41, 52, 53, 55, 57]

Code ID 62-21-6 · download JSON · raw on GitHub