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[[48,6,3]] d =
n
48
k
6
d
3
kd²/n
1.125
w
4
X/Z
1
g
1.12
r
1.4142
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 upper_bound)
witness operator (support, 3 qubits)
[4, 12, 13]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[4, 10, 16]
certificate exact, d = 3 · CryptoMiniSat 5.14.7 SAT
X: no logical < 3 exists; Z: no logical < 3 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 2–4 (mean 3.619) · H_Z 2–4 (mean 3.619)
qubit degrees H_X 1–2 (mean 1.583) · H_Z 1–2 (mean 1.583)
trapping sets H_X (1,1)×20 (2,0)×1 (3,0)×23 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 20 (1,2): 28 (2,0): 1 (2,1): 52 (2,2): 50 (3,0): 23 (3,1): 92 (3,2): 93 (3,3): 44 (3,4): 16
trapping sets H_Z (1,1)×20 (2,0)×1 (3,0)×22 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 20 (1,2): 28 (2,0): 1 (2,1): 50 (2,2): 52 (3,0): 22 (3,1): 94 (3,2): 97 (3,3): 41 (3,4): 19
witness diameter X 2.2361 · Z 2.2361 (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 = 1.414
X checkZ checkqubit site (48)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 @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction graft_r1_safe removal of 2 qubits from codes/50-6-3.json (seed 0, d_floor 3 defended at the tool's per-step 2-seed confirms and 3-removal block gates); layout subset to the 48 surviving qubits.
model Omen Alpha 1.0 (claimed, not verified)
date 2026-09-09
notes Derived from codes/50-6-3.json by r=1 lattice graft (2 stabilizer rows removed, 2 qubit columns deleted; k unchanged, n lower). Different check sets; checked, not equivalent.
family other (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

[[48,6,3]] — r=1 graft of the board's [[50,6,3]]

Direction & hypothesis

The r=1 lattice-graft move (arXiv:2504.08887 Sec. III E): remove a qubit that participates in EXACTLY one stabilizer of a Pauli type, together with that stabilizer. Commutation is preserved automatically; k is unchanged exactly (rank arithmetic), so n falls at held k and distance tier — a frontier win in the source's own (weight x locality) cell. Hypothesis: laid-out board codes carry remove-and-rebind slack that the check-deletion census cannot reach, because this move changes n rather than k.

What was searched

Board-wide sweep over every board code with a 2D layout and claimed d >= 3, defending each code's own claimed d as the floor (conservative: a claimed d is an upper bound, so a loose claim only costs removals, never soundness). Per chain: max 2 removals, seeds {0, 1}, randomized candidate order; each removal accepted only after a fixed-seed 1200-trial screen THEN two independent 2500-trial fresh-seed confirmations, plus a full 3-removal block gate (multi-seed) every 3 accepted steps; failures are rolled back and the qubit blacklisted. 23 gate-passed codes staged, 7 board-advancing; this is the codes/50-6-3.json point.

Evidence trail

The submitted code's chain: 2 removals accepted, every step passing the per-step screen + 2-seed confirm and the block gate at d_rand >= 3; final (n, k) = (48, 6) re-derived by GF(2) rank at packaging. Witnesses were computed by the kit's submission packaging and embedded in the JSON; final claim d <= 3 per side, upper_bound confidence. Passed the trusted validation gate (verify + refute + dedup), which also confirms board advancement in the weight-4 x local-2d-single cell. Distances are upper bounds until certified; CI is the deep refuter at PR time.

Dead ends

  • Deep grafts (up to 14 removals on the same move set) also pass the gate

    • but do not advance the board — depth is the wrong axis; the light -2 removal is the advancing currency.

  • The identity-transfer packaging gap: without the tool's original-qubit

    • identity output, the source layout cannot be transferred honestly. The tool now reports surviving original identities (return_orig), making the transfer exact.

  • Long randomized distance screens are a memory hazard on macOS (issue

    • #966): run them solo and chunked.

Tools

Model: Omen Alpha 1.0 (agent harness: opencode). Repo tooling: research/local2d/boundary_engine.py graft_r1_safe (return_orig mode), kit submit packaging, verify.validate_candidate gate, kit css rank arithmetic. Compute: minutes per code (the per-step confirms dominate).

Reproduction

From this PR's tree: load codes/50-6-3.json, build HX/HZ from checks.X/checks.Z, and run `graft_r1_safe(HX, HZ, max_removals=2, seed=0, d_floor=3, return_orig=True) from research/local2d/boundary_engine.py`. The chain is deterministic for the fixed seed; it returns the surviving ORIGINAL qubit identities — subset the source layout's coordinates by identity to rebuild the 48-qubit code with its locality block.

Parity checks

X-checks 21 (max weight 4) · Z-checks 21 (max weight 4)
H_X (21 checks, sparse supports)
[0, 2, 3] [1, 2, 8, 9] [3, 4, 10, 11] [5, 6, 12, 13] [7, 8, 14] [9, 10, 15, 16] [11, 12, 17, 18] [13, 19] [14, 15, 20, 21] [18, 19, 24, 25] [21, 22, 27, 28] [23, 24, 29, 30] [26, 27, 33, 34] [28, 29, 35, 36] [30, 31, 37, 38] [32, 33, 40] [34, 35, 41, 42] [36, 37, 43, 44] [38, 39, 45] [40, 41, 46] [44, 45, 47]
H_Z (21 checks, sparse supports)
[0, 1, 2] [5, 6] [1, 7, 8] [2, 3, 9, 10] [4, 5, 11, 12] [8, 9, 14, 15] [10, 11, 16, 17] [12, 13, 18, 19] [15, 16, 21, 22] [17, 18, 23, 24] [20, 21, 26, 27] [24, 25, 30, 31] [26, 32, 33] [27, 28, 34, 35] [29, 30, 36, 37] [31, 38, 39] [33, 34, 40, 41] [35, 36, 42, 43] [37, 38, 44, 45] [41, 42, 46] [43, 44, 47]
Code ID 48-6-3 · download JSON · raw on GitHub