[[98,6,12]] · QEC Challenge

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[[98,6,12]] d =

kd²/n

8.816

X/Z

0.0068

6.0

layers

swaps

533

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Distance

X/Z asymmetry 1 · d_X = 12, d_Z = 12 · 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 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[1, 4, 11, 16, 36, 46, 52, 59, 60, 64, 90, 93]

d_Z 12 · witness weight 12 (claimed upper_bound)

witness operator (support, 12 qubits)

[0, 21, 23, 34, 40, 60, 66, 79, 82, 85, 89, 95]

certificate exact, d = 12 · CryptoMiniSat 5.14 SAT

X: no logical < 12 exists (CryptoMiniSat, XOR + sequential-counter cardinality); Z: no logical < 12 exists (CryptoMiniSat, XOR + sequential-counter cardinality)

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)×98 (2,4)×735 (3,3)×98 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 98 (2,4): 735 (3,3): 98 (3,5): 7056 (3,7): 980

trapping sets H_Z (1,3)×98 (2,4)×735 (3,3)×98 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 98 (2,4): 735 (3,3): 98 (3,5): 7056 (3,7): 980

witness diameter X 7.2111 · Z 7.2111 (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)

Verified 2D layout

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

layout contributed by @dorakingx · Simulated annealing over site assignments on an integer grid with the repo's own research/local2d/fold_layout.search_layout (max-diameter objective, several bounding boxes and seeds); single layer at radius 4 tried first, then two layers at 7. Radius and per-site occupancy re-measured with the verifier's rule. · 2026-10-09

X checkZ checkqubit site (49)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 533 nearest-neighbor SWAPs per round in total, at most 11 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 Zijian Liang and Ke Liu and Hao Song and Yu-An Chen

provenance literature baseline

construction Twisted-torus bivariate-bicycle code from Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025). Stabilizers f(x,y)=1+x+x^-1y², g(x,y)=1+y+x^-2y^-1 on the abelian quotient group Z²/L with twist basis a_1=[0, 7], a_2=[7, 0] (n=2|det[a_1,a_2]|=2*49). Reconstructed from the published polynomials and twist; CSS form H_X=[f|g], H_Z=[gbar|fbar].

model classical construction (no AI model)

reference arXiv:2503.03827

date 2025

notes Reconstructed baseline from Liang, Liu, Song, Chen, Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025), arXiv:2503.03827. The [[n,k,d]] parameter set and stabilizer polynomials are published in that paper; this entry reproduces them. Distance is an upper bound per the paper (probabilistic for d>20); the witness is the surrogate logical of that weight. Seeded to fill a board coverage gap at this block size, not a discovery.

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 49 (max weight 6) · Z-checks 49 (max weight 6)

H_X (49 checks, sparse supports)

[0, 1, 34, 49, 51, 96] [1, 3, 5, 50, 53, 97] [2, 4, 39, 51, 54, 64] [3, 6, 8, 52, 56, 76] [4, 7, 9, 53, 57, 70] [5, 8, 43, 54, 58, 71] [6, 10, 12, 55, 60, 82] [7, 11, 13, 49, 56, 61] [8, 12, 14, 57, 62, 77] [9, 13, 46, 58, 63, 78] [10, 15, 17, 59, 65, 87] [11, 16, 18, 50, 60, 66] [12, 17, 19, 51, 61, 67] [13, 18, 20, 62, 68, 83] [14, 19, 48, 63, 69, 84] [15, 21, 23, 64, 71, 91] [16, 22, 24, 52, 65, 72] [17, 23, 25, 53, 66, 73] [18, 24, 26, 54, 67, 74] [19, 25, 27, 68, 75, 88] [20, 21, 26, 69, 76, 89] [0, 21, 29, 70, 77, 94] [22, 28, 30, 55, 71, 78] [23, 29, 31, 56, 72, 79] [24, 30, 32, 57, 73, 80] [25, 31, 33, 58, 74, 81] [0, 26, 32, 75, 82, 92] [27, 28, 33, 49, 76, 93] [2, 28, 35, 59, 77, 83] [29, 34, 36, 60, 78, 84] [30, 35, 37, 61, 79, 85] [31, 36, 38, 62, 80, 86] [1, 32, 37, 63, 81, 87] [2, 33, 38, 50, 82, 95] [5, 34, 40, 65, 83, 88] [35, 39, 41, 66, 84, 89] [36, 40, 42, 67, 85, 90] [3, 37, 41, 68, 86, 91] [4, 38, 42, 52, 69, 87] [9, 39, 44, 72, 88, 92] [40, 43, 45, 73, 89, 93] [6, 41, 44, 74, 90, 94] [7, 42, 45, 55, 75, 91] [14, 43, 47, 79, 92, 95] [10, 44, 46, 80, 93, 96] [11, 45, 47, 59, 81, 94] [15, 20, 46, 85, 95, 97] [16, 47, 48, 64, 86, 96] [22, 27, 48, 70, 90, 97]

H_Z (49 checks, sparse supports)

[0, 7, 27, 49, 70, 75] [1, 11, 33, 49, 50, 81] [0, 2, 12, 51, 77, 82] [3, 16, 38, 50, 52, 86] [1, 4, 17, 51, 53, 87] [2, 5, 18, 50, 54, 83] [6, 22, 42, 52, 55, 90] [3, 7, 23, 53, 56, 91] [4, 8, 24, 52, 54, 57] [5, 9, 25, 53, 58, 88] [10, 28, 45, 55, 59, 93] [6, 11, 29, 56, 60, 94] [7, 12, 30, 55, 57, 61] [8, 13, 31, 56, 58, 62] [9, 14, 32, 57, 63, 92] [2, 15, 47, 59, 64, 95] [10, 16, 34, 60, 65, 96] [11, 17, 35, 59, 61, 66] [12, 18, 36, 60, 62, 67] [13, 19, 37, 61, 63, 68] [14, 20, 38, 62, 69, 95] [4, 21, 48, 64, 69, 70] [5, 15, 22, 65, 71, 97] [16, 23, 39, 64, 66, 72] [17, 24, 40, 65, 67, 73] [18, 25, 41, 66, 68, 74] [19, 26, 42, 67, 69, 75] [3, 20, 27, 68, 76, 97] [8, 21, 28, 71, 76, 77] [9, 22, 29, 70, 72, 78] [23, 30, 43, 71, 73, 79] [24, 31, 44, 72, 74, 80] [25, 32, 45, 73, 75, 81] [6, 26, 33, 74, 76, 82] [13, 28, 34, 49, 78, 83] [14, 29, 35, 77, 79, 84] [30, 36, 46, 78, 80, 85] [31, 37, 47, 79, 81, 86] [10, 32, 38, 80, 82, 87] [19, 34, 39, 51, 84, 88] [20, 35, 40, 83, 85, 89] [36, 41, 48, 84, 86, 90] [15, 37, 42, 85, 87, 91] [26, 39, 43, 54, 89, 92] [27, 40, 44, 88, 90, 93] [21, 41, 45, 89, 91, 94] [33, 43, 46, 58, 93, 95] [0, 44, 47, 92, 94, 96] [1, 46, 48, 63, 96, 97]

Code ID 98-6-12 · download JSON · raw on GitHub