Distance
X/Z asymmetry 1 · d_X = 9, d_Z = 9 · 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 9 · witness weight 9 (claimed exact)
witness operator (support, 9 qubits)
[0, 1, 11, 21, 31, 41, 44, 51, 52]
d_Z 9 · witness weight 9 (claimed exact)
witness operator (support, 9 qubits)
[0, 9, 18, 27, 36, 45, 54, 64, 73]
certificate exact, d = 9 · scipy/HiGHS MILP
X: min over 1 Z-logical cosets = 9, all proven; Z: min over 1 X-logical cosets = 9, 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 2–4 (mean 3.6) · H_Z 2–4 (mean 3.6)
qubit degrees H_X 1–2 (mean 1.778) · H_Z 1–2 (mean 1.778)
trapping sets H_X (1,1)×18 (2,0)×8 (3,1)×94 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 18
(1,2): 63
(2,0): 8
(2,1): 34
(2,2): 158
(3,1): 94
(3,2): 405
(3,3): 16
(3,4): 96
trapping sets H_Z (1,1)×18 (2,0)×8 (3,1)×94 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 18
(1,2): 63
(2,0): 8
(2,1): 34
(2,2): 158
(3,1): 94
(3,2): 405
(3,3): 16
(3,4): 96
witness diameter X 8.9443 · Z 8.0623 (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/81-1-9/ · canonical noise recipe, 9 rounds, stim 1.16.0
d_circ ≤ 9 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 9 · fault-set witness of 9 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 9)
[5, 6, 31, 98, 100, 119, 140, 166, 466]
d_circ^Z 9 · fault-set witness of 9 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 9)
[12666, 12667, 12670, 12671, 12675, 12676, 12679, 12680, 12681]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see
Parity checks
X-checks 40 (max weight 4) · Z-checks 40 (max weight 4)
H_X (40 checks, sparse supports)
[0, 1, 9, 10]
[2, 3, 11, 12]
[4, 5, 13, 14]
[6, 7, 15, 16]
[8, 17]
[9, 18]
[10, 11, 19, 20]
[12, 13, 21, 22]
[14, 15, 23, 24]
[16, 17, 25, 26]
[18, 19, 27, 28]
[20, 21, 29, 30]
[22, 23, 31, 32]
[24, 25, 33, 34]
[26, 35]
[27, 36]
[28, 29, 37, 38]
[30, 31, 39, 40]
[32, 33, 41, 42]
[34, 35, 43, 44]
[36, 37, 45, 46]
[38, 39, 47, 48]
[40, 41, 49, 50]
[42, 43, 51, 52]
[44, 53]
[45, 54]
[46, 47, 55, 56]
[48, 49, 57, 58]
[50, 51, 59, 60]
[52, 53, 61, 62]
[54, 55, 63, 64]
[56, 57, 65, 66]
[58, 59, 67, 68]
[60, 61, 69, 70]
[62, 71]
[63, 72]
[64, 65, 73, 74]
[66, 67, 75, 76]
[68, 69, 77, 78]
[70, 71, 79, 80]
H_Z (40 checks, sparse supports)
[0, 1]
[2, 3]
[4, 5]
[6, 7]
[1, 2, 10, 11]
[3, 4, 12, 13]
[5, 6, 14, 15]
[7, 8, 16, 17]
[9, 10, 18, 19]
[11, 12, 20, 21]
[13, 14, 22, 23]
[15, 16, 24, 25]
[19, 20, 28, 29]
[21, 22, 30, 31]
[23, 24, 32, 33]
[25, 26, 34, 35]
[27, 28, 36, 37]
[29, 30, 38, 39]
[31, 32, 40, 41]
[33, 34, 42, 43]
[37, 38, 46, 47]
[39, 40, 48, 49]
[41, 42, 50, 51]
[43, 44, 52, 53]
[45, 46, 54, 55]
[47, 48, 56, 57]
[49, 50, 58, 59]
[51, 52, 60, 61]
[55, 56, 64, 65]
[57, 58, 66, 67]
[59, 60, 68, 69]
[61, 62, 70, 71]
[63, 64, 72, 73]
[65, 66, 74, 75]
[67, 68, 76, 77]
[69, 70, 78, 79]
[73, 74]
[75, 76]
[77, 78]
[79, 80]