[[72,12,6]] · QEC Challenge

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[[72,12,6]] d =

kd²/n

6.0

X/Z

0.0136

4.5826

layers

swaps

243

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Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 6, w_Z = 6 (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 operator (support, 6 qubits)

[48, 52, 66, 67, 68, 71]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness operator (support, 6 qubits)

[2, 16, 24, 40, 48, 62]

certificate exact, d = 6 · scipy/HiGHS MILP

X: no logical < 6 exists; Z: no logical < 6 exists

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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 6

qubit degrees H_X 3 · H_Z 3

trapping sets H_X (1,3)×72 (2,4)×540 (3,3)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 72 (2,4): 540 (3,3): 72 (3,5): 5184 (3,7): 720

trapping sets H_Z (1,3)×72 (2,4)×540 (3,3)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 72 (2,4): 540 (3,3): 72 (3,5): 5184 (3,7): 720

witness diameter X 5.0 · Z 4.3589 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Circuit tier

syndrome-extraction memory circuits committed under circuits/72-12-6/ · canonical noise recipe, 6 rounds, stim 1.16.0

d_circ ≤ 6 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 6 · fault-set witness of 6 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 6)

[13950, 13990, 14418, 14477, 14799, 15390]

d_circ^Z 6 · fault-set witness of 6 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 6)

[13892, 13897, 15056, 15087, 15843, 15873]

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

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

X checkZ checkqubit site (39)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

routing cost 243 nearest-neighbor SWAPs per round in total, at most 7 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors Bravyi, Sergey and Cross, Andrew W. and Gambetta, Jay M. and Maslov, Dmitri and Rall, Patrick and Yoder, Theodore J.

provenance literature baseline

construction Bivariate bicycle code, orders (6, 6), A=x³ + y² + y, B=x² + x + y³ (periodic boundary conditions).

model classical construction (no AI model)

reference arXiv:2308.07915

date 2023-08-15

notes Distance d=6 established in arXiv:2308.07915 (Table 3); witness here is a decoder-found logical operator of that weight (upper bound). 2D-local layout added 2026-07-27: simulated annealing over qubit-to-site assignments on a unit-spaced triangular grid (2 layers, capacity 1 per site per layer); measured interaction radius 4.582576 (bilayer); same code, same check matrices.

family bivariate bicycle (a tag, not a ranking)

locality 2D-local bilayer (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 36 (max weight 6) · Z-checks 36 (max weight 6)

H_X (36 checks, sparse supports)

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

H_Z (36 checks, sparse supports)

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

Code ID 72-12-6 · download JSON · raw on GitHub