Distance
X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed exact)
witness operator (support, 8 qubits)
[0, 8, 17, 26, 35, 43, 50, 57]
d_Z 8 · witness weight 8 (claimed exact)
witness operator (support, 8 qubits)
[6, 7, 56, 57, 58, 59, 60, 61]
certificate exact, d = 8 · scipy/HiGHS MILP
X: min over 2 Z-logical cosets = 8, all proven; Z: min over 2 X-logical cosets = 8, 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)×64 (2,2)×192 (3,2)×576 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 64
(2,2): 192
(3,2): 576
(3,4): 128
trapping sets H_Z (1,2)×64 (2,2)×192 (3,2)×576 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 64
(2,2): 192
(3,2): 576
(3,4): 128
witness diameter X 9.2195 · Z 7.0 (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)
Circuit tier
syndrome-extraction memory circuits committed under
circuits/64-2-8/ · canonical noise recipe, 8 rounds, stim 1.16.0
d_circ ≤ 8 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 8 · fault-set witness of 8 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 8)
[840, 848, 984, 1010, 1700, 1701, 2373, 2419]
d_circ^Z 8 · fault-set witness of 8 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 8)
[10580, 10581, 10586, 10589, 10594, 10599, 10600, 10606]
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 8x8 torus, distance 8.
model classical construction (no AI model)
date 1997-07-09
notes Distance is exact by construction. 2D layout added 2026-07-23: the 8x8 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 32 (max weight 4) · Z-checks 32 (max weight 4)
H_X (32 checks, sparse supports)
[0, 7, 56, 63]
[1, 2, 57, 58]
[3, 4, 59, 60]
[5, 6, 61, 62]
[0, 1, 8, 9]
[2, 3, 10, 11]
[4, 5, 12, 13]
[6, 7, 14, 15]
[8, 15, 16, 23]
[9, 10, 17, 18]
[11, 12, 19, 20]
[13, 14, 21, 22]
[16, 17, 24, 25]
[18, 19, 26, 27]
[20, 21, 28, 29]
[22, 23, 30, 31]
[24, 31, 32, 39]
[25, 26, 33, 34]
[27, 28, 35, 36]
[29, 30, 37, 38]
[32, 33, 40, 41]
[34, 35, 42, 43]
[36, 37, 44, 45]
[38, 39, 46, 47]
[40, 47, 48, 55]
[41, 42, 49, 50]
[43, 44, 51, 52]
[45, 46, 53, 54]
[48, 49, 56, 57]
[50, 51, 58, 59]
[52, 53, 60, 61]
[54, 55, 62, 63]
H_Z (32 checks, sparse supports)
[0, 1, 56, 57]
[2, 3, 58, 59]
[4, 5, 60, 61]
[6, 7, 62, 63]
[0, 7, 8, 15]
[1, 2, 9, 10]
[3, 4, 11, 12]
[5, 6, 13, 14]
[8, 9, 16, 17]
[10, 11, 18, 19]
[12, 13, 20, 21]
[14, 15, 22, 23]
[16, 23, 24, 31]
[17, 18, 25, 26]
[19, 20, 27, 28]
[21, 22, 29, 30]
[24, 25, 32, 33]
[26, 27, 34, 35]
[28, 29, 36, 37]
[30, 31, 38, 39]
[32, 39, 40, 47]
[33, 34, 41, 42]
[35, 36, 43, 44]
[37, 38, 45, 46]
[40, 41, 48, 49]
[42, 43, 50, 51]
[44, 45, 52, 53]
[46, 47, 54, 55]
[48, 55, 56, 63]
[49, 50, 57, 58]
[51, 52, 59, 60]
[53, 54, 61, 62]