X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 4, w_Z = 4 (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 exact)
witness operator (support, 4 qubits)
[0, 4, 8, 12]
d_Z 4 · witness weight 4 (claimed exact)
witness operator (support, 4 qubits)
[2, 7, 8, 13]
certificateexact, d = 4 · scipy/HiGHS MILP
X: min over 2 Z-logical cosets = 4, all proven; Z: min over 2 X-logical cosets = 4, all proven
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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 4 · H_Z 4
qubit degrees H_X 2 · H_Z 2
trapping sets H_X (1,2)×16 (2,2)×48 (3,2)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 16
(2,2): 48
(3,2): 144
(3,4): 32
trapping sets H_Z (1,2)×16 (2,2)×48 (3,2)×144 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 16
(2,2): 48
(3,2): 144
(3,4): 32
witness diameter X 3.0 · Z 4.2426 (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)
d_circ ≤ 4 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 4 · fault-set witness of 4 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 4)
[820, 822, 975, 977]
d_circ^Z 4 · fault-set witness of 4 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 4)
[1165, 1166, 1176, 1177]
ler/round (X) 0.00173 95% CI [0.00157, 0.00191] · 413/6×104 shots · decoder bposd-cs-10 at p=0.001
ler/round (Z) 0.00171 95% CI [0.00155, 0.00188] · 408/6×104 shots · decoder bposd-cs-10 at p=0.001
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 (16)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 20 nearest-neighbor SWAPs per round in total, at most 3 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 Kitaev, A. Yu.
provenance literature baseline
construction Rotated toric code on an 4x4 torus, distance 4.
notes Distance is exact by construction. 2D layout added 2026-07-23: the 4x4 torus folded into the plane (ring fold i -> 2i / 2(L-i)-1 per axis), making the wrap-around checks geometrically local; single layer, measured interaction radius 2*sqrt(2).
family topological (a tag, not a ranking)
locality 2D-local single (computed from the layout)
weight class weight ≤ 4 (computed)
How this code was found
no research note was staged with this submission — notes are requested for new submissions (notes/README.md)