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[[7,1,3]] d =
n
7
k
1
d
3
kd²/n
1.286
w
4
X/Z
1
g
0.321
r
2.0
layers
1
swaps
0

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (claimed exact)
witness operator (support, 3 qubits)
[1, 3, 5]
d_Z 3 · witness weight 3 (claimed exact)
witness operator (support, 3 qubits)
[2, 4, 5]
certificate exact, d = 3 · scipy/HiGHS MILP
X: min over 1 Z-logical cosets = 3, all proven; Z: min over 1 X-logical cosets = 3, 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 4 · H_Z 4 (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 1–3 (mean 1.714) · H_Z 1–3 (mean 1.714)
trapping sets H_X (1,1)×3 (2,1)×9 (3,0)×7 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 3 (1,2): 3 (1,3): 1 (2,1): 9 (2,2): 6 (3,0): 7 (3,1): 12 (3,2): 6 (3,3): 3
trapping sets H_Z (1,1)×3 (2,1)×9 (3,0)×7 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 3 (1,2): 3 (1,3): 1 (2,1): 9 (2,2): 6 (3,0): 7 (3,1): 12 (3,2): 6 (3,3): 3
witness diameter X 1.7321 · 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)

Circuit tier

syndrome-extraction memory circuits committed under circuits/7-1-3/ · canonical noise recipe, 3 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)
witness fault set (mechanism indices in the committed .dem, 2)
[47, 78]
d_circ^Z 2 · fault-set witness of 2 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 2)
[269, 270]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see
transversal gates 1 verified (qubit permutation, uniform Clifford, or block-to-block CX; action checked over GF(2) modulo stabilizers)
logical Hadamard H on every qubit · X0 → Z0, Z0 → X0
logical basis the gate actions refer to
X0 = [0, 1, 2, 3, 4, 5, 6]; Z0 = [0, 1, 2, 3, 4, 5, 6]

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 2
check (X = Z, self-dual)qubit site (7)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 0 nearest-neighbor SWAPs per round in total, at most 0 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 Steane, Andrew M.
provenance literature baseline
construction Steane code: the distance-3 2D color code.
model classical construction (no AI model)
reference arXiv:quant-ph/9601029
date 1996-01-29
notes Distance is exact by construction. 2D layout added 2026-07-23: compact unit-spaced triangular patch (qubit 6, shared by all three faces, at the centre); single layer, measured interaction radius 2.0.
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)

Parity checks

X-checks 3 (max weight 4) · Z-checks 3 (max weight 4)
H_X (3 checks, sparse supports)
[3, 4, 5, 6] [1, 2, 5, 6] [0, 2, 4, 6]
H_Z (3 checks, sparse supports)
[3, 4, 5, 6] [1, 2, 5, 6] [0, 2, 4, 6]
Code ID 7-1-3 · download JSON · raw on GitHub