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[[63,2,7]] d =
n
63
k
2
d
7
kd²/n
1.556
w
4
X/Z
1.14
g
0.0972
r
2.8284
layers
1
swaps
127

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Distance

X/Z asymmetry 1.14 · d_X = 8, d_Z = 7 · 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 upper_bound)
witness operator (support, 8 qubits)
[5, 14, 22, 29, 37, 45, 51, 59]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[0, 8, 23, 30, 38, 47, 55]
certificate exact, d = 7 · scipy/HiGHS MILP
X: no logical < 8 exists; Z: no logical < 7 exists

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 3–4 (mean 3.938)
qubit degrees H_X 1–2 (mean 1.905) · H_Z 2
trapping sets H_X (1,1)×6 (2,1)×18 (3,0)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 6 (1,2): 57 (2,1): 18 (2,2): 162 (3,0): 2 (3,1): 50 (3,2): 461 (3,3): 18 (3,4): 102
trapping sets H_Z (1,2)×63 (2,2)×186 (3,2)×549 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 63 (2,2): 186 (3,2): 549 (3,4): 122
witness diameter X 7.0 · Z 7.6158 (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)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 2.828
X checkZ checkqubit site (63)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 127 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 @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Check deletion + degree-one r=1 graft of the [[64,2,8]] toric code
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-17
notes Checked against the board: no exact-duplicate or WL-equivalent entry (trusted gate).
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

the research note submitted with this code · raw markdown · all notes

[[63,2,7]] — check-deletion + r=1 graft of the [[64,2,8]] toric code

Direction & hypothesis

Issue #1155 hackathon, 2D-local priority. Composition hypothesis: deleting a check (raising k) can create a degree-one qubit that a subsequent r=1 graft removes, producing a (n−1, k+Δk, d') point neither move reaches alone.

What was searched

Source: [[64,2,8]] (toric code, single-layer 2D-local layout, family topological). Move: delete one check, then one degree-one r=1 graft. Original column identities tracked so the source coordinates subset honestly; layers preserved.

Evidence trail

Trusted gate (verify/validate_candidate, refute=True): passed=true, board_advancing=true in the weight-4 × local-2d-single cell, no exact duplicate / WL-equivalent board entry, not refuted. Fresh witness search at submission: d ≤ 7, upper bound. Not an exact certificate; literature novelty unverified.

Dead ends

Two high-k single-layer leads ([[453,8,18]], [[452,8,17]]) were refuted at the gate and excluded. Plain deletion-control and incident-deletion-control moves produced no board-advancing survivors on their own.

Tools

Model: DeepSeek V4 Flash 0731 (Zed agent). Repo tooling: research/kit (make_submission, css GF(2) exports), verify/validate_candidate (trusted gate), verify/qldpc_verify (structural screening + board reports). Local compute only.

Reproduction

Rebuild (H_X, H_Z) from the source [[64,2,8]] toric-code matrix: 1. Delete one check row (the deleted row is recorded in the submission's provenance trail; the move raises k by the rank loss). 2. Graft a degree-one qubit: pick a column q that participates in exactly one stabilizer of some Pauli type, remove that column and its unique same-side row, and truncate opposite-type rows containing q (commutation is preserved by construction). 3. Subset the source's 2D coordinates to the surviving columns (original identities, not guessed) and keep the single-layer layout.

The method is the r=1 lattice grafting of arXiv:2504.08887 Sec. III E combined with check deletion; the exact deleted row/column identities are deterministic from the source and the move order described above. The packaged JSON carries embedded witnesses and seed for the distance claim.

Parity checks

X-checks 30 (max weight 4) · Z-checks 32 (max weight 4)
H_X (30 checks, sparse supports)
[0, 7, 55, 62] [1, 2, 56, 57] [3, 4, 58, 59] [5, 6, 60, 61] [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] [41, 42, 48, 49] [43, 44, 50, 51] [45, 46, 52, 53] [49, 50, 57, 58] [51, 52, 59, 60] [53, 54, 61, 62]
H_Z (32 checks, sparse supports)
[0, 1, 55, 56] [2, 3, 57, 58] [4, 5, 59, 60] [6, 7, 61, 62] [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] [42, 43, 49, 50] [44, 45, 51, 52] [46, 47, 53, 54] [54, 55, 62] [48, 49, 56, 57] [50, 51, 58, 59] [52, 53, 60, 61]
Code ID 63-2-7 · download JSON · raw on GitHub