[[368,8,8]] · QEC Challenge

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[[368,8,8]] d =

368

kd²/n

1.391

X/Z

0.087

2.8284

layers

swaps

904

Share this result

Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 exact)

witness operator (support, 8 qubits)

[0, 1, 2, 3, 4, 5, 22, 23]

d_Z 8 · witness weight 8 (claimed exact)

witness operator (support, 8 qubits)

[238, 239, 257, 260, 276, 289, 306, 307]

certificate exact, d = 8 · scipy/HiGHS MILP

X: no logical < 8 exists; Z: no logical < 8 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 3–4 (mean 3.909) · H_Z 2–4 (mean 3.826)

qubit degrees H_X 1–2 (mean 1.87) · H_Z 1–2 (mean 1.913)

trapping sets H_X (1,1)×48 (2,0)×16 (3,1)×336 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 48 (1,2): 320 (2,0): 16 (2,1): 112 (2,2): 880 (3,1): 336 (3,2): 2432 (3,3): 80 (3,4): 544

trapping sets H_Z (1,1)×32 (2,1)×96 (3,0)×16 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 32 (1,2): 336 (2,1): 96 (2,2): 928 (3,0): 16 (3,1): 256 (3,2): 2560 (3,3): 96 (3,4): 576

witness diameter X 14.1421 · Z 8.2462 (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 (368)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 904 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 @FarLab

provenance submitted through the challenge

novelty novelty not audited

construction Dual-grid hole packing of the L=20 rotated toric code (checkerboard 2x2-cell checks, one qubit per site): four smooth (X-boundary) 2x2 holes on a pitch-10 grid and four rough (Z-boundary) 2x2 holes on the dual grid offset (5,5); hole boundaries completed to a maximal commuting set of 2x2-box checks. Each hole carries one logical qubit (k=8); d=8 = min(hole loop 4h, inter-hole string). Member of the [[23h²m², 2m², 4h]] family, which sits at kd²/(c² n) = 2/23 at every h and m -- above the 1/16 of the planar surface code at equal capacity. 2D layout: both torus axes ring-folded into the plane (i -> 2i / 2(L-i)-1 per axis, as for the board's toric baselines); single layer, measured interaction radius 2*sqrt(2).

model Claude Fable 5 (claimed, not verified)

date 2026-08-13

notes Distance exact by two independent methods: connectivity-graph shortest-nontrivial-path/cycle search, and scipy/HiGHS MILP certificate (no logical < 8 exists, both sides; certs/368-8-8.json). Construction and verification scripts: research/holes/ (branch research/hole-packing-constant).

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

[[368,8,8]] — dual-grid holey rotated toric code (family g → 32/23 at the r=√2 floor)

Direction & hypothesis

Hole-encoded logical qubits (defect encoding) are textbook for surface codes, and the folklore says they are inefficient: a hole pays its perimeter (loop ≥ d) *and* its exclusion zone (strings ≥ d), capping single-type packings at kd²/(c²n) = 1/24 — below the plain surface code. Hypothesis, following the fixed-d hole-packing observation of the merged [[700,85,3]] (PR #464): the folklore misses that (i) string weights on the rotated lattice follow the Chebyshev metric (the connectivity graphs are diagonally-adjacent grids), and (ii) smooth and rough holes barely constrain each other (no light "lasso" mode down to gap ≈ h). Together these let two same-type square packings interleave at full density.

Construction

L=20 rotated toric code (checkerboard 2×2-cell checks, one qubit per site). Four smooth (X-boundary) 2×2 holes at pitch 10; four rough (Z-boundary) 2×2 holes on the dual grid, offset (5,5). Hole boundaries = greedy completion to a maximal commuting set of 2×2-box checks (weights 2–4). Each hole carries one logical qubit: k = 8. Distance 8 = min(hole loop 4h = 8, same-type inter-hole string 10−2 = 8); the mixed gap (Chebyshev 3) sits above the lasso threshold.

Member of the exact family [[23h²m², 2m², 4h]] (h even, any m). In board normalization: this folded toric instance scores g = 4kd²/(nρ²r⁴) = 0.0870 at r = 2√2 (torus folding pays the standard 16× in g, as for the board's toric baselines); the *planar open-boundary windows* of the same pattern live at the r = √2 packing floor and approach g = 32/23 ≈ 1.391 > 1 as the window grows — 39% above the surface code's g = 1.0, with d → ∞ rather than at fixed d. Equivalently, in the capacity normalization of the sharp-constants note, the family sits at kd²/(c²n) = 2/23 > 1/16. Planar windows keep d = 8 exactly (verified to n = 5085), so the efficiency is not a wraparound artifact.

Verification

verify/qldpc_verify.py: ok — CSS commutes, k = 8 recomputed, weight ≤ 4,

local-2d-single, interaction radius 2√2, single layer.

verify/validate_candidate.py: passed, board_advancing: true

(weight-4 × local-2d-single), no exact/WL duplicate, refute gate finds no lighter logical in 8000 RIS trials.

Exact distance, two independent methods (certs/368-8-8.json):

scipy/HiGHS MILP — "no logical < 8 exists", both sides; and a connectivity-graph exact search (shortest nontrivial path/cycle over all logical classes). Both return d_X = d_Z = 8.

Weight-8 witnesses for both sides included in the code file.

Context

The [[23h²m², 2m², 4h]] family and its calibration (loop/string/lasso laws, measured exactly at h = 2,3,4) are documented in research/holes/NOTE.md on branch research/hole-packing-constant, with constructors and both distance engines. Defect encoding itself is standard (e.g. Raussendorf–Harrington, Fowler et al., arXiv:1208.0928); the contribution here is the two-type dual-grid packing at density 2/R² and the resulting constant 2/23 > 1/16, which bears on the sharp-constant question for the BPT tradeoff at fixed capacity.

Parity checks

X-checks 176 (max weight 4) · Z-checks 184 (max weight 4)

H_X (176 checks, sparse supports)

[0, 1, 16, 17] [2, 3, 18, 19] [4, 5, 20, 21] [6, 7, 22, 23] [8, 9, 24, 25] [10, 11, 26, 27] [12, 13, 28, 29] [14, 15, 30, 31] [17, 18, 35, 36] [19, 20, 37, 38] [21, 22, 39, 40] [25, 26, 45, 46] [27, 28, 47, 48] [29, 30, 49, 50] [32, 33, 52, 53] [34, 35, 54, 55] [36, 37, 56, 57] [38, 39, 58, 59] [40, 41, 60, 61] [42, 43, 62, 63] [44, 45, 64, 65] [46, 47, 66, 67] [48, 49, 68, 69] [50, 51, 70, 71] [53, 54, 73, 74] [55, 56, 75, 76] [57, 58, 77, 78] [59, 60, 79, 80] [61, 62, 81, 82] [63, 64, 83, 84] [65, 66, 85, 86] [67, 68, 87, 88] [69, 70, 89, 90] [52, 71, 72, 91] [72, 73, 92, 93] [74, 75, 94, 95] [80, 81, 98, 99] [82, 83, 100, 101] [84, 85, 102, 103] [90, 91, 106, 107] [93, 94, 109, 110] [95, 96, 111, 112] [97, 98, 113, 114] [99, 100, 115, 116] [101, 102, 117, 118] [103, 104, 119, 120] [105, 106, 121, 122] [92, 107, 108, 123] [108, 109, 124, 125] [110, 111, 126, 127] [114, 115, 132, 133] [116, 117, 134, 135] [118, 119, 136, 137] [122, 123, 142, 143] [125, 126, 145, 146] [127, 128, 147, 148] [129, 130, 149, 150] [131, 132, 151, 152] [133, 134, 153, 154] [135, 136, 155, 156] [137, 138, 157, 158] [139, 140, 159, 160] [141, 142, 161, 162] [124, 143, 144, 163] [144, 145, 164, 165] [146, 147, 166, 167] [148, 149, 168, 169] [150, 151, 170, 171] [152, 153, 172, 173] [154, 155, 174, 175] [156, 157, 176, 177] [158, 159, 178, 179] [160, 161, 180, 181] [162, 163, 182, 183] [167, 168, 185, 186] [169, 170, 187, 188] [171, 172, 189, 190] [177, 178, 193, 194] [179, 180, 195, 196] [181, 182, 197, 198] [184, 185, 200, 201] [186, 187, 202, 203] [188, 189, 204, 205] [190, 191, 206, 207] [192, 193, 208, 209] [194, 195, 210, 211] [196, 197, 212, 213] [198, 199, 214, 215] [201, 202, 219, 220] [203, 204, 221, 222] [205, 206, 223, 224] [209, 210, 229, 230] [211, 212, 231, 232] [213, 214, 233, 234] [216, 217, 236, 237] [218, 219, 238, 239] [220, 221, 240, 241] [222, 223, 242, 243] [224, 225, 244, 245] [226, 227, 246, 247] [228, 229, 248, 249] [230, 231, 250, 251] [232, 233, 252, 253] [234, 235, 254, 255] [237, 238, 257, 258] [239, 240, 259, 260] [241, 242, 261, 262] [243, 244, 263, 264] [245, 246, 265, 266] [247, 248, 267, 268] [249, 250, 269, 270] [251, 252, 271, 272] [253, 254, 273, 274] [236, 255, 256, 275] [256, 257, 276, 277] [258, 259, 278, 279] [264, 265, 282, 283] [266, 267, 284, 285] [268, 269, 286, 287] [274, 275, 290, 291] [277, 278, 293, 294] [279, 280, 295, 296] [281, 282, 297, 298] [283, 284, 299, 300] [285, 286, 301, 302] [287, 288, 303, 304] [289, 290, 305, 306] [276, 291, 292, 307] [292, 293, 308, 309] [294, 295, 310, 311] [298, 299, 316, 317] [300, 301, 318, 319] [302, 303, 320, 321] [306, 307, 326, 327] [309, 310, 329, 330] [311, 312, 331, 332] [313, 314, 333, 334] [315, 316, 335, 336] [317, 318, 337, 338] [319, 320, 339, 340] [321, 322, 341, 342] [323, 324, 343, 344] [325, 326, 345, 346] [308, 327, 328, 347] [328, 329, 348, 349] [330, 331, 350, 351] [332, 333, 352, 353] [334, 335, 354, 355] [336, 337, 356, 357] [338, 339, 358, 359] [340, 341, 360, 361] [342, 343, 362, 363] [344, 345, 364, 365] [346, 347, 366, 367] [1, 2, 351, 352] [3, 4, 353, 354] [5, 6, 355, 356] [9, 10, 361, 362] [11, 12, 363, 364] [13, 14, 365, 366] [16, 33, 34] [23, 41, 42] [24, 43, 44] [31, 32, 51] [165, 166, 184] [173, 174, 191] [175, 176, 192] [164, 183, 199] [200, 217, 218] [207, 225, 226] [208, 227, 228] [215, 216, 235] [0, 349, 350] [7, 357, 358] [8, 359, 360] [15, 348, 367]

H_Z (184 checks, sparse supports)

[1, 2, 17, 18] [3, 4, 19, 20] [5, 6, 21, 22] [9, 10, 25, 26] [11, 12, 27, 28] [13, 14, 29, 30] [16, 17, 34, 35] [18, 19, 36, 37] [20, 21, 38, 39] [22, 23, 40, 41] [24, 25, 44, 45] [26, 27, 46, 47] [28, 29, 48, 49] [30, 31, 50, 51] [33, 34, 53, 54] [35, 36, 55, 56] [37, 38, 57, 58] [39, 40, 59, 60] [41, 42, 61, 62] [43, 44, 63, 64] [45, 46, 65, 66] [47, 48, 67, 68] [49, 50, 69, 70] [32, 51, 52, 71] [52, 53, 72, 73] [54, 55, 74, 75] [56, 57, 76, 77] [58, 59, 78, 79] [60, 61, 80, 81] [62, 63, 82, 83] [64, 65, 84, 85] [66, 67, 86, 87] [68, 69, 88, 89] [70, 71, 90, 91] [73, 74, 93, 94] [75, 76, 95, 96] [79, 80, 97, 98] [81, 82, 99, 100] [83, 84, 101, 102] [85, 86, 103, 104] [89, 90, 105, 106] [72, 91, 92, 107] [92, 93, 108, 109] [94, 95, 110, 111] [98, 99, 114, 115] [100, 101, 116, 117] [102, 103, 118, 119] [106, 107, 122, 123] [109, 110, 125, 126] [111, 112, 127, 128] [113, 114, 131, 132] [115, 116, 133, 134] [117, 118, 135, 136] [119, 120, 137, 138] [121, 122, 141, 142] [108, 123, 124, 143] [124, 125, 144, 145] [126, 127, 146, 147] [128, 129, 148, 149] [130, 131, 150, 151] [132, 133, 152, 153] [134, 135, 154, 155] [136, 137, 156, 157] [138, 139, 158, 159] [140, 141, 160, 161] [142, 143, 162, 163] [145, 146, 165, 166] [147, 148, 167, 168] [149, 150, 169, 170] [151, 152, 171, 172] [153, 154, 173, 174] [155, 156, 175, 176] [157, 158, 177, 178] [159, 160, 179, 180] [161, 162, 181, 182] [144, 163, 164, 183] [166, 167, 184, 185] [168, 169, 186, 187] [170, 171, 188, 189] [172, 173, 190, 191] [176, 177, 192, 193] [178, 179, 194, 195] [180, 181, 196, 197] [182, 183, 198, 199] [185, 186, 201, 202] [187, 188, 203, 204] [189, 190, 205, 206] [193, 194, 209, 210] [195, 196, 211, 212] [197, 198, 213, 214] [200, 201, 218, 219] [202, 203, 220, 221] [204, 205, 222, 223] [206, 207, 224, 225] [208, 209, 228, 229] [210, 211, 230, 231] [212, 213, 232, 233] [214, 215, 234, 235] [217, 218, 237, 238] [219, 220, 239, 240] [221, 222, 241, 242] [223, 224, 243, 244] [225, 226, 245, 246] [227, 228, 247, 248] [229, 230, 249, 250] [231, 232, 251, 252] [233, 234, 253, 254] [216, 235, 236, 255] [236, 237, 256, 257] [238, 239, 258, 259] [240, 241, 260, 261] [242, 243, 262, 263] [244, 245, 264, 265] [246, 247, 266, 267] [248, 249, 268, 269] [250, 251, 270, 271] [252, 253, 272, 273] [254, 255, 274, 275] [257, 258, 277, 278] [259, 260, 279, 280] [263, 264, 281, 282] [265, 266, 283, 284] [267, 268, 285, 286] [269, 270, 287, 288] [273, 274, 289, 290] [256, 275, 276, 291] [276, 277, 292, 293] [278, 279, 294, 295] [282, 283, 298, 299] [284, 285, 300, 301] [286, 287, 302, 303] [290, 291, 306, 307] [293, 294, 309, 310] [295, 296, 311, 312] [297, 298, 315, 316] [299, 300, 317, 318] [301, 302, 319, 320] [303, 304, 321, 322] [305, 306, 325, 326] [292, 307, 308, 327] [308, 309, 328, 329] [310, 311, 330, 331] [312, 313, 332, 333] [314, 315, 334, 335] [316, 317, 336, 337] [318, 319, 338, 339] [320, 321, 340, 341] [322, 323, 342, 343] [324, 325, 344, 345] [326, 327, 346, 347] [329, 330, 349, 350] [331, 332, 351, 352] [333, 334, 353, 354] [335, 336, 355, 356] [337, 338, 357, 358] [339, 340, 359, 360] [341, 342, 361, 362] [343, 344, 363, 364] [345, 346, 365, 366] [328, 347, 348, 367] [0, 1, 350, 351] [2, 3, 352, 353] [4, 5, 354, 355] [6, 7, 356, 357] [8, 9, 360, 361] [10, 11, 362, 363] [12, 13, 364, 365] [14, 15, 366, 367] [77, 78] [87, 88] [96, 112] [97, 113] [104, 120] [105, 121] [129, 130] [139, 140] [261, 262] [271, 272] [280, 296] [281, 297] [288, 304] [289, 305] [313, 314] [323, 324]

Code ID 368-8-8 · download JSON · raw on GitHub