[[897,173,3]] · QEC Challenge

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[[897,173,3]] d =

897

173

kd²/n

1.736

X/Z

0.0174

4.4721

layers

swaps

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 8, w_Z = 8 (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)

[532, 593, 624]

d_Z 3 · witness weight 3 (claimed exact)

witness operator (support, 3 qubits)

[7, 29, 51]

certificate exact, d = 3 · CryptoMiniSat 5.14 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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 2–8 (mean 3.947) · H_Z 2–8 (mean 3.85)

qubit degrees H_X 1–3 (mean 1.575) · H_Z 1–3 (mean 1.571)

trapping sets H_X (1,1)×384 (2,0)×29 (3,0)×607 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 384 (1,2): 510 (1,3): 3 (2,0): 29 (2,1): 1095 (2,2): 1028 (2,3): 10 (3,0): 607 (3,1): 2208 (3,2): 2099 (3,3): 1144 (3,4): 430 (3,5): 4

trapping sets H_Z (1,1)×387 (2,0)×22 (3,0)×566 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 387 (1,2): 508 (1,3): 2 (2,0): 22 (2,1): 1066 (2,2): 984 (2,3): 7 (3,0): 566 (3,1): 2054 (3,2): 1955 (3,3): 1039 (3,4): 356 (3,5): 2

witness diameter X 4.2426 · 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

X checkZ checkqubit site (897)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 70 nearest-neighbor SWAPs per round in total, at most 5 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 @vprusso

provenance submitted through the challenge

novelty novelty not audited

construction Affine mod-5 checkerboard rectangle completion on 32x31, phases (1,4), order 3, projected to the encoding core, contracted by 85 boundary-first pivot moves of degree at most 3 (seed 0) with exact d>=3 and single-block acceptance (research/geometric_grafts/campaign.py)

model Claude Fable 5.1 (claimed, not verified)

date 2026-10-12

family topological (a tag, not a ranking)

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

weight class weight ≤ 8 (computed)

How this code was found

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

[[897,173,3]] contracted affine checkerboard core, exact d = 3

Direction & hypothesis

Issue #2939 recorded a contraction sweep of the affine mod-5 checkerboard cores that was killed at 11 of 36 seeds while still descending, and named [[897,173,3]] from base 32x31, phases (1,4), order 3, seed 0 as the first candidate: it beats codes/899-173-3.json (#2938) on n at equal k and d in the weight-8 x local-2d-bilayer cell. This entry is that chain, reproduced, and the completed sweep around it.

Construction and tooling are #2938's (research/geometric_grafts/campaign.py, @MathysRennela): rectangle completion of the affine mod-5 checkerboard grammar of notes/961-169-3.md, projection onto the encoding core, then pivot moves of degree at most 3 (XOR an own-side row into every other own-side row covering a qubit, delete the row, delete the qubit) accepted only when k is unchanged, d >= 3 holds exactly, and the stabilizer group stays one connected block. Our part is running the sweep to completion and filing what it found.

What was searched

All 36 planned chains, one process per (base, seed), on a laptop and on a Perlmutter CPU node (genesis campaign tooling; every chain also saves its artifact, so a kill loses nothing). The sixth base of #2939's list, 34x29 at order 0, is incomplete in the constructor and yields no chain, as campaign.py itself reports; the other five bases ran six seeds each.

| base | k | n reached by seed 0..5 | best | |---|---:|---|---:| | 34x29 p=(3,1) o=3 | 173 | 899 x 3 (#2938; six of six in that run) | 899 | | 32x31 p=(1,4) o=3 | 173 | 897, 898, 897, 897, 897, 899 | 897 | | 31x31 p=(3,1) o=3 | 169 | 869, 871, 875, 869, 873, 871 | 869 | | 36x26 p=(3,1) o=3 | 164 | 844, 842, 840, 838, 840, 840 | 838 | | 36x27 p=(2,2) o=2 | 170 | 890, 890, 888, 890, 888, 890 | 888 |

Four of six seeds on this base end at exactly 897, and none below, so the base has converged rather than been sampled lightly. A chain costs 745 to 950 s on 8 threads (85 accepted moves here; 8 to 11 s per accepted move, dominated by the two RREFs each candidate move needs).

The chain: base 32x31, phases (1,4), order 3 gives the completion; projected onto its encoding core, n = 982, k = 173; 85 accepted moves, seed 0, deterministic, n = 897. Pivot moves leave a few empty or repeated check rows behind (four on the X side here); those are dropped when the matrices are written out, which changes neither the stabilizer group nor the code.

Evidence trail

Lower bound, d >= 3: the campaign's exact test (no weight-1 or weight-2

logical, decided by column enumeration against the opposite checks plus rowspace membership), re-run on the final matrices.

Upper bound, d <= 3: weight-3 witnesses on both sides in the JSON, found

by the submit pipeline's RIS and re-checked by the gate.

Certificate: certs/897-173-3.json, level proof_log. verify/sat_certify.py

encodes "a nontrivial logical of weight <= 2 exists" per side; kissat 4.0.4 reports UNSAT on both and drat-trim verifies both refutations; the XOR path (pycryptosat) agrees. The entry claims confidence: exact on both sides, which the verifier accepts as an upper bound until the certificate is read.

Gate (verify/validate_candidate.py against main at 9de3faad): passed, no

exact or WL duplicate, refuter found nothing lighter, board_advancing true, advances_by: ["n"], cell weight-8 x local-2d-bilayer.

Where it sits

Max check weight 8 and max check diameter sqrt(20) = 4.47 after contraction, so the cell is weight-8 x local-2d-bilayer, as for #2938. There, k = 173 at d = 3 was held by [[899,173,3]]; this entry reaches it at n = 897 and dominates that one on n alone. The weight-4 records [[923,173,3]] and [[925,173,3]] are untouched: contraction cannot hold w <= 4 (#2939, "What is dead").

Dead ends

The weight-4 x local-2d-single cell stays out of reach by this move set:

the capped variant (constrained.py) stalls near n = 950 at k = 173 whether or not degree-2 and degree-3 moves are allowed, exactly as #2939 measured.

More seeds on this base do not help: four of six already end at 897.

Tools

research/geometric_grafts/campaign.py (#2938) driven one chain per process by genesis campaign scripts; verify/sat_certify.py with kissat and drat-trim for the certificate; ./qldpc submit for the witnesses, layout, and verification. Claude Fable 5.1 (Claude Code) working for @vprusso.

Reproduction

import sys, numpy as np
sys.path[:0] = ["research/geometric_grafts", "research/geometric_grafts/874-157-3",
                "research/kit", "verify", "research/local2d"]
from campaign import base_state, graft_campaign, one_connected, exact_ge_3

HX, HZ, coords, orig, k0 = base_state(32, 31, 1, 4, 3)   # n=982, k=173
a, b, c, o, rm = graft_campaign(HX, HZ, coords, list(orig), k0=k0, seed=0,
                                rng=np.random.default_rng(0))
assert a.shape[1] == 897 and rm == 85 and exact_ge_3(a, b) and one_connected(a, b)[0]

Drop empty and repeated rows of a and b; the remaining rows are this entry's checks in order, and c its coordinates.

Parity checks

X-checks 358 (max weight 8) · Z-checks 366 (max weight 8)

H_X (358 checks, sparse supports)

[0, 1, 22, 23] [4, 26] [5, 6, 27, 28] [7, 8, 29, 30] [11, 33] [12, 13, 34, 35] [14, 15, 36, 37] [18, 40] [23, 24, 43, 44] [3, 25, 45, 46] [26, 27, 49, 50] [28, 29, 51, 52] [30, 31, 53, 54] [10, 32, 55, 56] [33, 34, 59, 60] [35, 36, 61, 62] [37, 38, 63, 64] [19, 20, 41, 68] [44, 45, 70, 71] [46, 47, 72, 73] [48, 49, 74, 75] [52, 53, 78, 79] [54, 55, 80, 81] [56, 57, 82, 83] [58, 59, 84, 85] [62, 63, 88, 89] [64, 65, 90, 91] [66, 67, 92, 93] [71, 72, 98, 99] [73, 74, 100, 101] [75, 76, 102, 103] [77, 78, 104, 105] [81, 82, 108, 109] [83, 84, 110, 111] [85, 86, 112, 113] [87, 88, 114, 115] [91, 92, 118, 119] [40, 41, 93, 94, 120, 121] [94, 95, 122, 123] [97, 98, 126, 127] [101, 102, 130, 131] [103, 104, 132, 133] [105, 106, 134, 135] [107, 108, 136, 137] [111, 112, 140, 141] [113, 114, 142, 143] [115, 116, 144, 145] [117, 118, 146, 147] [121, 122, 150, 151] [127, 128, 155, 156] [129, 130, 157, 158] [133, 134, 161, 162] [135, 136, 163, 164] [137, 138, 165, 166] [139, 140, 167, 168] [143, 144, 171, 172] [145, 146, 173, 174] [147, 148, 175, 176] [149, 150, 177, 178] [153, 154, 181, 182] [156, 157, 185, 186] [158, 159, 187, 188] [160, 161, 189, 190] [164, 165, 193, 194] [166, 167, 195, 196] [168, 169, 197, 198] [170, 171, 199, 200] [174, 175, 203, 204] [176, 177, 205, 206] [178, 179, 207, 208] [180, 181, 209, 210] [186, 187, 216, 217] [188, 189, 218, 219] [190, 191, 220, 221] [192, 193, 222, 223] [196, 197, 226, 227] [198, 199, 228, 229] [200, 201, 230, 231] [202, 203, 232, 233] [206, 207, 236, 237] [210, 211, 239, 240] [212, 213, 241, 242] [125, 126, 155, 184, 213, 214, 243, 244] [215, 216, 245, 246] [219, 220, 249, 250] [221, 222, 251, 252] [223, 224, 253, 254] [225, 226, 255, 256] [229, 230, 259, 260] [231, 232, 261, 262] [233, 234, 263, 264] [235, 236, 265, 266] [242, 243, 271, 272] [244, 245, 273, 274] [246, 247, 275, 276] [248, 249, 277, 278] [252, 253, 281, 282] [254, 255, 283, 284] [256, 257, 285, 286] [258, 259, 287, 288] [262, 263, 291, 292] [264, 265, 293, 294] [266, 267, 295, 296] [270, 271, 298, 299] [274, 275, 302, 303] [276, 277, 304, 305] [278, 279, 306, 307] [280, 281, 308, 309] [284, 285, 312, 313] [286, 287, 314, 315] [288, 289, 316, 317] [290, 291, 318, 319] [294, 295, 322, 323] [296, 297, 324, 325] [208, 209, 238, 239, 268, 297, 326, 327] [299, 300, 330, 331] [301, 302, 332, 333] [305, 306, 336, 337] [307, 308, 338, 339] [309, 310, 340, 341] [311, 312, 342, 343] [315, 316, 346, 347] [317, 318, 348, 349] [319, 320, 350, 351] [321, 322, 352, 353] [325, 326, 356, 357] [327, 328, 358, 359] [331, 332, 361, 362] [333, 334, 363, 364] [335, 336, 365, 366] [339, 340, 369, 370] [341, 342, 371, 372] [343, 344, 373, 374] [345, 346, 375, 376] [349, 350, 379, 380] [351, 352, 381, 382] [353, 354, 383, 384] [355, 356, 385, 386] [362, 363, 390, 391] [364, 365, 392, 393] [366, 367, 394, 395] [368, 369, 396, 397] [372, 373, 400, 401] [374, 375, 402, 403] [376, 377, 404, 405] [378, 379, 406, 407] [382, 383, 410, 411] [384, 385, 412, 413] [386, 387, 414, 415] [389, 390, 419, 420] [393, 394, 423, 424] [395, 396, 425, 426] [397, 398, 427, 428] [399, 400, 429, 430] [403, 404, 433, 434] [405, 406, 435, 436] [407, 408, 437, 438] [409, 410, 439, 440] [413, 414, 443, 444] [415, 416, 445, 446] [420, 421, 449, 450] [422, 423, 451, 452] [426, 427, 455, 456] [428, 429, 457, 458] [430, 431, 459, 460] [432, 433, 461, 462] [436, 437, 465, 466] [438, 439, 467, 468] [440, 441, 469, 470] [442, 443, 471, 472] [446, 447, 475, 476] [450, 451, 479, 480] [452, 453, 481, 482] [454, 455, 483, 484] [458, 459, 487, 488] [460, 461, 489, 490] [462, 463, 491, 492] [464, 465, 493, 494] [468, 469, 497, 498] [470, 471, 499, 500] [472, 473, 501, 502] [474, 475, 503, 504] [480, 481, 510, 511] [482, 483, 512, 513] [484, 485, 514, 515] [486, 487, 516, 517] [490, 491, 520, 521] [492, 493, 522, 523] [494, 495, 524, 525] [496, 497, 526, 527] [500, 501, 530, 531] [504, 505, 533, 534] [506, 507, 535, 536] [418, 419, 449, 478, 507, 508, 537, 538] [509, 510, 539, 540] [513, 514, 543, 544] [515, 516, 545, 546] [517, 518, 547, 548] [519, 520, 549, 550] [523, 524, 553, 554] [525, 526, 555, 556] [527, 528, 557, 558] [529, 530, 559, 560] [502, 503, 532, 533, 563] [536, 537, 566, 567] [538, 539, 568, 569] [540, 541, 570, 571] [542, 543, 572, 573] [546, 547, 576, 577] [548, 549, 578, 579] [550, 551, 580, 581] [552, 553, 582, 583] [556, 557, 586, 587] [558, 559, 588, 589] [560, 561, 590, 591] [565, 566, 594, 595] [569, 570, 598, 599] [571, 572, 600, 601] [573, 574, 602, 603] [575, 576, 604, 605] [579, 580, 608, 609] [581, 582, 610, 611] [583, 584, 612, 613] [585, 586, 614, 615] [589, 590, 618, 619] [591, 592, 620, 621] [562, 592, 593, 622, 623] [595, 596, 626, 627] [597, 598, 628, 629] [601, 602, 632, 633] [603, 604, 634, 635] [605, 606, 636, 637] [607, 608, 638, 639] [611, 612, 642, 643] [613, 614, 644, 645] [615, 616, 646, 647] [617, 618, 648, 649] [621, 622, 652, 653] [627, 628, 655, 656] [629, 630, 657, 658] [631, 632, 659, 660] [635, 636, 663, 664] [637, 638, 665, 666] [639, 640, 667, 668] [641, 642, 669, 670] [645, 646, 673, 674] [647, 648, 675, 676] [649, 650, 677, 678] [651, 652, 679, 680] [658, 659, 684, 685] [660, 661, 686, 687] [662, 663, 688, 689] [666, 667, 692, 693] [668, 669, 694, 695] [670, 671, 696, 697] [672, 673, 698, 699] [676, 677, 702, 703] [678, 679, 704, 705] [685, 686, 712, 713] [687, 688, 714, 715] [689, 690, 716, 717] [691, 692, 718, 719] [695, 696, 722, 723] [697, 698, 724, 725] [699, 700, 726, 727] [701, 702, 728, 729] [705, 732, 733] [680, 681, 706, 734] [656, 657, 683, 708, 709, 736, 737] [709, 710, 738, 739] [711, 712, 740, 741] [715, 716, 744, 745] [717, 718, 746, 747] [719, 720, 748, 749] [721, 722, 750, 751] [725, 726, 754, 755] [727, 728, 756, 757] [729, 730, 758, 759] [731, 732, 760, 761] [734, 764, 765] [739, 740, 769, 770] [741, 742, 771, 772] [743, 744, 773, 774] [747, 748, 777, 778] [749, 750, 779, 780] [751, 752, 781, 782] [753, 754, 783, 784] [757, 758, 787, 788] [759, 760, 789, 790] [761, 762, 791, 792] [763, 764, 793] [770, 771, 799, 800] [772, 773, 801, 802] [774, 775, 803, 804] [776, 777, 805, 806] [780, 781, 809, 810] [782, 783, 811, 812] [784, 785, 813, 814] [786, 787, 815, 816] [790, 791, 819, 820] [792, 821] [793, 794, 822, 823] [795, 796, 824, 825] [737, 738, 767, 768, 796, 797, 826, 827] [798, 799, 828, 829] [802, 803, 832, 833] [804, 805, 834, 835] [806, 807, 836, 837] [808, 809, 838, 839] [812, 813, 842, 843] [814, 815, 844, 845] [816, 817, 846, 847] [818, 819, 848, 849] [822, 852, 853] [827, 828, 856, 857] [829, 830, 858, 859] [831, 832, 860, 861] [837, 838, 863, 864] [839, 840, 865, 866] [841, 842, 867, 868] [847, 848, 870, 871] [849, 850, 872, 873] [851, 852, 874, 875] [857, 858, 878, 879] [859, 860, 880, 881] [864, 865, 885, 886] [866, 867, 887, 888] [868, 889] [871, 872, 892, 893] [873, 874, 894, 895] [875, 896] [15, 16] [891, 892] [20, 21] [884, 885] [17, 39, 65, 66] [1, 2] [877, 878] [825, 826, 855, 876] [823, 854] [8, 9] [42, 43] [835, 836, 861, 862, 882, 883] [416, 417] [68, 123, 152, 153] [845, 846, 869, 890] [329, 330] [835, 836, 862, 883] [623, 624, 654, 682, 706, 707] [706, 707] [625, 626, 708, 735] [69, 96, 124, 125] [360, 388, 418, 448] [124, 183] [269, 328] [448, 477] [563, 564, 593, 624] [735, 736, 766, 767]

H_Z (366 checks, sparse supports)

[1, 2, 23, 24] [3, 25] [6, 7, 28, 29] [8, 9, 30, 31] [10, 32] [13, 14, 35, 36] [15, 16, 37, 38] [17, 39] [22, 23, 42, 43] [24, 25, 44, 45] [27, 28, 50, 51] [29, 30, 52, 53] [31, 32, 54, 55] [11, 33, 58, 59] [34, 35, 60, 61] [36, 37, 62, 63] [38, 39, 64, 65] [45, 46, 71, 72] [47, 48, 73, 74] [49, 50, 75, 76] [51, 52, 77, 78] [55, 56, 81, 82] [57, 58, 83, 84] [59, 60, 85, 86] [61, 62, 87, 88] [65, 66, 91, 92] [18, 40, 67, 93] [69, 96] [70, 71, 97, 98] [72, 73, 99, 100] [74, 75, 101, 102] [78, 79, 105, 106] [80, 81, 107, 108] [82, 83, 109, 110] [84, 85, 111, 112] [88, 89, 115, 116] [90, 91, 117, 118] [92, 93, 119, 120] [94, 121, 122] [96, 97, 125, 126] [98, 99, 127, 128] [102, 103, 131, 132] [104, 105, 133, 134] [106, 107, 135, 136] [108, 109, 137, 138] [112, 113, 141, 142] [114, 115, 143, 144] [116, 117, 145, 146] [118, 119, 147, 148] [122, 123, 151, 152] [128, 129, 156, 157] [130, 131, 158, 159] [132, 133, 160, 161] [134, 135, 162, 163] [138, 139, 166, 167] [140, 141, 168, 169] [142, 143, 170, 171] [144, 145, 172, 173] [148, 149, 176, 177] [150, 151, 178, 179] [152, 153, 180, 181] [155, 156, 184, 185] [157, 158, 186, 187] [159, 160, 188, 189] [163, 164, 192, 193] [165, 166, 194, 195] [167, 168, 196, 197] [169, 170, 198, 199] [173, 174, 202, 203] [175, 176, 204, 205] [177, 178, 206, 207] [179, 180, 208, 209] [124, 125, 183, 212, 213] [184, 214] [185, 186, 215, 216] [189, 190, 219, 220] [191, 192, 221, 222] [193, 194, 223, 224] [195, 196, 225, 226] [199, 200, 229, 230] [201, 202, 231, 232] [203, 204, 233, 234] [205, 206, 235, 236] [209, 210, 239] [213, 242, 243] [216, 217, 246, 247] [218, 219, 248, 249] [220, 221, 250, 251] [222, 223, 252, 253] [226, 227, 256, 257] [228, 229, 258, 259] [230, 231, 260, 261] [232, 233, 262, 263] [236, 237, 266, 267] [238, 268] [243, 244, 272, 273] [245, 246, 274, 275] [247, 248, 276, 277] [249, 250, 278, 279] [253, 254, 282, 283] [255, 256, 284, 285] [257, 258, 286, 287] [259, 260, 288, 289] [263, 264, 292, 293] [265, 266, 294, 295] [267, 268, 296, 297] [271, 272, 299, 300] [273, 274, 301, 302] [275, 276, 303, 304] [279, 280, 307, 308] [281, 282, 309, 310] [283, 284, 311, 312] [285, 286, 313, 314] [289, 290, 317, 318] [291, 292, 319, 320] [293, 294, 321, 322] [295, 296, 323, 324] [239, 240, 269, 327, 328] [298, 299, 329, 330] [300, 301, 331, 332] [304, 305, 335, 336] [306, 307, 337, 338] [308, 309, 339, 340] [310, 311, 341, 342] [314, 315, 345, 346] [316, 317, 347, 348] [318, 319, 349, 350] [320, 321, 351, 352] [324, 325, 355, 356] [326, 327, 357, 358] [332, 333, 362, 363] [334, 335, 364, 365] [336, 337, 366, 367] [338, 339, 368, 369] [342, 343, 372, 373] [344, 345, 374, 375] [346, 347, 376, 377] [348, 349, 378, 379] [352, 353, 382, 383] [354, 355, 384, 385] [356, 357, 386, 387] [358, 359] [360, 388] [361, 362, 389, 390] [363, 364, 391, 392] [365, 366, 393, 394] [369, 370, 397, 398] [371, 372, 399, 400] [373, 374, 401, 402] [375, 376, 403, 404] [379, 380, 407, 408] [381, 382, 409, 410] [383, 384, 411, 412] [385, 386, 413, 414] [388, 389, 418, 419] [390, 391, 420, 421] [394, 395, 424, 425] [396, 397, 426, 427] [398, 399, 428, 429] [400, 401, 430, 431] [404, 405, 434, 435] [406, 407, 436, 437] [408, 409, 438, 439] [410, 411, 440, 441] [414, 415, 444, 445] [416, 417, 446, 447] [421, 422, 450, 451] [423, 424, 452, 453] [425, 426, 454, 455] [427, 428, 456, 457] [431, 432, 460, 461] [433, 434, 462, 463] [435, 436, 464, 465] [437, 438, 466, 467] [441, 442, 470, 471] [443, 444, 472, 473] [445, 446, 474, 475] [449, 450, 478, 479] [451, 452, 480, 481] [453, 454, 482, 483] [457, 458, 486, 487] [459, 460, 488, 489] [461, 462, 490, 491] [463, 464, 492, 493] [467, 468, 496, 497] [469, 470, 498, 499] [471, 472, 500, 501] [473, 474, 502, 503] [418, 448, 477, 506, 507] [478, 508] [479, 480, 509, 510] [483, 484, 513, 514] [485, 486, 515, 516] [487, 488, 517, 518] [489, 490, 519, 520] [493, 494, 523, 524] [495, 496, 525, 526] [497, 498, 527, 528] [499, 500, 529, 530] [503, 504, 533] [507, 536, 537] [510, 511, 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Code ID 897-173-3 · download JSON · raw on GitHub