X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)

d_Z 2 · witness weight 2 (claimed upper_bound)

certificate exact, d = 2 · scipy/HiGHS MILP

X: no logical < 2 exists; Z: no logical < 2 exists

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 6 · H_Z 6

qubit degrees H_X 3 · H_Z 3

trapping sets H_X (1,3)×12 (2,0)×6 (3,3)×208 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

trapping sets H_Z (1,3)×12 (2,0)×6 (3,3)×208 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

witness diameter X 1.0 · Z 2.0 (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)

syndrome-extraction memory circuits committed under circuits/12-4-2/ · 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)

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

ler/round (X) 0.0253 95% CI [0.0231, 0.0276] · 493/10⁴ shots · decoder bposd-cs-10 at p=0.001

ler/round (Z) 0.0257 95% CI [0.0236, 0.0281] · 501/10⁴ shots · decoder bposd-cs-10 at p=0.001

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

routing cost 2 nearest-neighbor SWAPs per round in total, at most 1 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

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+xy, g(x,y)=1+y+xy on the abelian quotient group Z²/L with twist basis a_1=[0, 3], a_2=[2, 1] (n=2|det[a_1,a_2]|=2*6). 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. 2D-local layout added 2026-07-27: simulated annealing over qubit-to-site assignments on a unit-spaced triangular grid (1 layer, capacity 1 per site per layer); measured interaction radius 2.645751 (single-layer); same code, same check matrices.

family bivariate bicycle (a tag, not a ranking)

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

weight class weight ≤ 6 (computed)

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

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

Code ID 12-4-2 · download JSON · raw on GitHub