Distance
X/Z asymmetry 1 · d_X = 6, d_Z = 6 · 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 6 · witness weight 6 (claimed exact)
witness operator (support, 6 qubits)
[3, 9, 14, 20, 26, 32]
d_Z 6 · witness weight 6 (claimed exact)
witness operator (support, 6 qubits)
[0, 7, 14, 21, 28, 35]
certificate exact, d = 6 · scipy/HiGHS MILP
X: min over 2 Z-logical cosets = 6, all proven; Z: min over 2 X-logical cosets = 6, 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)×36 (2,2)×108 (3,2)×324 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 36
(2,2): 108
(3,2): 324
(3,4): 72
trapping sets H_Z (1,2)×36 (2,2)×108 (3,2)×324 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 36
(2,2): 108
(3,2): 324
(3,4): 72
witness diameter X 5.099 · Z 7.0711 (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/36-2-6/ · 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)
[14, 16, 26, 41, 52, 72]
d_circ^Z 6 · fault-set witness of 6 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 6)
[4284, 4286, 4297, 4306, 4308, 4314]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see
Construction & provenance
authors Kitaev, A. Yu.
provenance literature baseline
construction Rotated toric code on an 6x6 torus, distance 6.
model classical construction (no AI model)
date 1997-07-09
notes Distance is exact by construction. 2D layout added 2026-07-23: the 6x6 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)
Parity checks
X-checks 18 (max weight 4) · Z-checks 18 (max weight 4)
H_X (18 checks, sparse supports)
[0, 5, 30, 35]
[1, 2, 31, 32]
[3, 4, 33, 34]
[0, 1, 6, 7]
[2, 3, 8, 9]
[4, 5, 10, 11]
[6, 11, 12, 17]
[7, 8, 13, 14]
[9, 10, 15, 16]
[12, 13, 18, 19]
[14, 15, 20, 21]
[16, 17, 22, 23]
[18, 23, 24, 29]
[19, 20, 25, 26]
[21, 22, 27, 28]
[24, 25, 30, 31]
[26, 27, 32, 33]
[28, 29, 34, 35]
H_Z (18 checks, sparse supports)
[0, 1, 30, 31]
[2, 3, 32, 33]
[4, 5, 34, 35]
[0, 5, 6, 11]
[1, 2, 7, 8]
[3, 4, 9, 10]
[6, 7, 12, 13]
[8, 9, 14, 15]
[10, 11, 16, 17]
[12, 17, 18, 23]
[13, 14, 19, 20]
[15, 16, 21, 22]
[18, 19, 24, 25]
[20, 21, 26, 27]
[22, 23, 28, 29]
[24, 29, 30, 35]
[25, 26, 31, 32]
[27, 28, 33, 34]