[[198,8,9]] · QEC Challenge

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[[198,8,9]] d =

198

kd²/n

3.273

X/Z

0.0511

2.8284

layers

swaps

161

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Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · w_X = 6, w_Z = 6 (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 upper_bound)

witness operator (support, 9 qubits)

[11, 12, 42, 63, 64, 84, 86, 112, 120]

d_Z 9 · witness weight 9 (claimed upper_bound)

witness operator (support, 9 qubits)

[58, 76, 78, 80, 86, 90, 93, 179, 190]

certificate exact, d = 9 · CryptoMiniSat 5.14.7 SAT

X: no logical < 9 exists; Z: no logical < 9 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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 2–6 (mean 4.958) · H_Z 2–6 (mean 5.011)

qubit degrees H_X 1–3 (mean 2.404) · H_Z 1–3 (mean 2.379)

trapping sets H_X (1,1)×40 (2,0)×8 (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): 40 (1,2): 38 (1,3): 120 (2,0): 8 (2,1): 48 (2,2): 159 (2,3): 193 (2,4): 645 (3,0): 23 (3,1): 97 (3,2): 370 (3,3): 1199 (3,4): 1740 (3,5): 4701 (3,6): 174 (3,7): 713

trapping sets H_Z (1,1)×42 (2,0)×11 (3,0)×20 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,1): 42 (1,2): 39 (1,3): 117 (2,0): 11 (2,1): 53 (2,2): 162 (2,3): 196 (2,4): 624 (3,0): 20 (3,1): 112 (3,2): 378 (3,3): 1234 (3,4): 1756 (3,5): 4526 (3,6): 185 (3,7): 681

witness diameter X 8.6023 · Z 8.544 (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

X checkZ checkqubit site (100)2 qubits stacked (2 layers)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 161 nearest-neighbor SWAPs per round in total, at most 2 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/200-8-9.json (seed 0, d_floor 9 defended at the tool's per-step 2-seed confirms and 3-removal block gates); layout subset to the 198 surviving qubits.

model Omen Alpha 1.0 (claimed, not verified)

date 2026-09-09

notes Derived from codes/200-8-9.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 bivariate bicycle (a tag, not a ranking)

locality 2D-local bilayer (computed from the layout)

weight class weight ≤ 6 (computed)

How this code was found

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

[[198,8,9]] — r=1 graft of the board's [[200,8,9]]

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/200-8-9.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 >= 9; final (n, k) = (198, 8) 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 <= 9 per side, upper_bound confidence. Passed the trusted validation gate (verify + refute + dedup), which also confirms board advancement in the weight-6 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/200-8-9.json, build HX/HZ from checks.X/checks.Z, and run `graft_r1_safe(HX, HZ, max_removals=2, seed=0, d_floor=9, 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 198-qubit code with its locality block.

Parity checks

X-checks 96 (max weight 6) · Z-checks 94 (max weight 6)

H_X (96 checks, sparse supports)

[0, 100, 101] [1, 101, 102] [2, 102, 103] [3, 103, 104] [4, 104, 105] [5, 105, 106] [6, 106] [7, 107] [0, 10, 109, 110] [1, 11, 110, 111] [2, 12, 111, 112] [3, 13, 112, 113] [4, 14, 113, 114] [5, 15, 114, 115] [6, 16, 115, 116] [7, 17, 116, 117] [2, 10, 20, 99, 119, 120] [3, 11, 21, 100, 120, 121] [4, 12, 22, 101, 121, 122] [5, 13, 23, 102, 122, 123] [6, 14, 24, 103, 123, 124] [7, 15, 25, 104, 124, 125] [8, 16, 26, 105, 125, 126] [9, 17, 27, 106, 126, 127] [12, 20, 30, 108, 129, 130] [13, 21, 31, 109, 130, 131] [14, 22, 32, 110, 131, 132] [15, 23, 33, 111, 132, 133] [16, 24, 34, 112, 133, 134] [17, 25, 35, 113, 134, 135] [18, 26, 36, 114, 135, 136] [19, 27, 37, 115, 136, 137] [22, 30, 40, 118, 139, 140] [23, 31, 41, 119, 140, 141] [24, 32, 42, 120, 141, 142] [25, 33, 43, 121, 142, 143] [26, 34, 44, 122, 143, 144] [27, 35, 45, 123, 144, 145] [28, 36, 46, 124, 145, 146] [29, 37, 47, 125, 146, 147] [32, 40, 50, 128, 149, 150] [33, 41, 51, 129, 150, 151] [34, 42, 52, 130, 151, 152] [35, 43, 53, 131, 152, 153] [36, 44, 54, 132, 153, 154] [37, 45, 55, 133, 154, 155] [38, 46, 56, 134, 155, 156] [39, 47, 57, 135, 156, 157] [42, 50, 60, 138, 159, 160] [43, 51, 61, 139, 160, 161] [44, 52, 62, 140, 161, 162] [45, 53, 63, 141, 162, 163] [46, 54, 64, 142, 163, 164] [47, 55, 65, 143, 164, 165] [48, 56, 66, 144, 165, 166] [49, 57, 67, 145, 166, 167] [52, 60, 70, 148, 169, 170] [53, 61, 71, 149, 170, 171] [54, 62, 72, 150, 171, 172] [55, 63, 73, 151, 172, 173] [56, 64, 74, 152, 173, 174] [57, 65, 75, 153, 174, 175] [58, 66, 76, 154, 175, 176] [59, 67, 77, 155, 176, 177] [62, 70, 80, 158, 179, 180] [63, 71, 81, 159, 180, 181] [64, 72, 82, 160, 181, 182] [65, 73, 83, 161, 182, 183] [66, 74, 84, 162, 183, 184] [67, 75, 85, 163, 184, 185] [68, 76, 86, 164, 185, 186] [69, 77, 87, 165, 186, 187] [72, 80, 168, 189, 190] [73, 81, 90, 169, 190, 191] [74, 82, 91, 170, 191, 192] [75, 83, 92, 171, 192, 193] [76, 84, 93, 172, 193, 194] [77, 85, 94, 173, 194, 195] [78, 86, 95, 174, 195, 196] [79, 87, 96, 175, 196, 197] [82, 178] [83, 90, 179] [84, 91, 180] [85, 92, 181] [86, 93, 182] [87, 94, 183] [88, 95, 184] [89, 96, 185] [91, 188] [92, 189] [93, 190] [94, 191] [95, 192] [96, 193] [97, 194] [98, 195]

H_Z (94 checks, sparse supports)

[20, 99, 108] [0, 21, 100, 109] [0, 1, 22, 101, 110, 118] [1, 2, 23, 102, 111, 119] [2, 3, 24, 103, 112, 120] [3, 4, 25, 104, 113, 121] [4, 5, 26, 105, 114, 122] [5, 6, 27, 106, 115, 123] [7, 8, 29, 107, 117, 125] [8, 9, 126] [9, 127] [30, 108, 118] [10, 31, 109, 119] [10, 11, 32, 110, 120, 128] [11, 12, 33, 111, 121, 129] [12, 13, 34, 112, 122, 130] [13, 14, 35, 113, 123, 131] [14, 15, 36, 114, 124, 132] [15, 16, 37, 115, 125, 133] [16, 17, 38, 116, 126, 134] [17, 18, 39, 117, 127, 135] [18, 19, 136] [19, 137] [40, 118, 128] [20, 41, 119, 129] [20, 21, 42, 120, 130, 138] [21, 22, 43, 121, 131, 139] [22, 23, 44, 122, 132, 140] [23, 24, 45, 123, 133, 141] [24, 25, 46, 124, 134, 142] [25, 26, 47, 125, 135, 143] [26, 27, 48, 126, 136, 144] [27, 28, 49, 127, 137, 145] [28, 29, 146] [29, 147] [50, 128, 138] [30, 51, 129, 139] [30, 31, 52, 130, 140, 148] [31, 32, 53, 131, 141, 149] [32, 33, 54, 132, 142, 150] [33, 34, 55, 133, 143, 151] [34, 35, 56, 134, 144, 152] [35, 36, 57, 135, 145, 153] [36, 37, 58, 136, 146, 154] [37, 38, 59, 137, 147, 155] [38, 39, 156] [39, 157] [60, 138, 148] [40, 61, 139, 149] [40, 41, 62, 140, 150, 158] [41, 42, 63, 141, 151, 159] [42, 43, 64, 142, 152, 160] [43, 44, 65, 143, 153, 161] [44, 45, 66, 144, 154, 162] [45, 46, 67, 145, 155, 163] [46, 47, 68, 146, 156, 164] [47, 48, 69, 147, 157, 165] [48, 49, 166] [49, 167] [70, 148, 158] [50, 71, 149, 159] [50, 51, 72, 150, 160, 168] [51, 52, 73, 151, 161, 169] [52, 53, 74, 152, 162, 170] [53, 54, 75, 153, 163, 171] [54, 55, 76, 154, 164, 172] [55, 56, 77, 155, 165, 173] [56, 57, 78, 156, 166, 174] [57, 58, 79, 157, 167, 175] [58, 59, 176] [59, 177] [80, 158, 168] [60, 81, 159, 169] [60, 61, 82, 160, 170, 178] [61, 62, 83, 161, 171, 179] [62, 63, 84, 162, 172, 180] [63, 64, 85, 163, 173, 181] [64, 65, 86, 164, 174, 182] [65, 66, 87, 165, 175, 183] [66, 67, 88, 166, 176, 184] [67, 68, 89, 167, 177, 185] [68, 69, 186] [69, 187] [70, 90, 169, 179] [70, 71, 91, 170, 180, 188] [71, 72, 92, 171, 181, 189] [72, 73, 93, 172, 182, 190] [73, 74, 94, 173, 183, 191] [74, 75, 95, 174, 184, 192] [75, 76, 96, 175, 185, 193] [76, 77, 97, 176, 186, 194] [77, 78, 98, 177, 187, 195] [78, 79, 196] [79, 197]

Code ID 198-8-9 · download JSON · raw on GitHub