[[125,6,4]] · QEC Challenge

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[[125,6,4]] d =

125

kd²/n

0.768

X/Z

1.25

0.768

1.4142

layers

swaps

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Distance

X/Z asymmetry 1.25 · d_X = 5, d_Z = 4 · 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 5 · witness weight 5 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02

witness operator (support, 5 qubits)

[120, 121, 122, 123, 124]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02

witness operator (support, 4 qubits)

[0, 11, 12, 13]

certificate exact, d = 4 · graphlike_A (source sha256 66a15ffb) and graphlike_B (56d04349), two independent exact programs for graph-like check matrices; github.com/TimoKellerMath/quantum-error-correction @ ae52261bfdbd, board-certificates-2026-11 · level solver: a solver verdict, no artifact a third party can check without re-running the encoding

X: graphlike_A and graphlike_B: d_X = 5, the length of a shortest homologically nontrivial cycle of the graph of H_Z (no nontrivial X-logical of weight <= 4); words of weight 5 rechecked; Z: graphlike_A and graphlike_B: d_Z = 4, the length of a shortest homologically nontrivial cycle of the graph of H_X (no nontrivial Z-logical of weight <= 3); words of weight 4 rechecked

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 3.576) · H_Z 2–4 (mean 3.533)

qubit degrees H_X 1–2 (mean 1.688) · H_Z 1–2 (mean 1.696)

trapping sets H_X (1,1)×39 (2,0)×12 (3,0)×5 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 39 (1,2): 86 (2,0): 12 (2,1): 83 (2,2): 195 (3,0): 5 (3,1): 210 (3,2): 454 (3,3): 54 (3,4): 105

trapping sets H_Z (1,1)×38 (2,0)×11 (3,0)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 38 (1,2): 87 (2,0): 11 (2,1): 79 (2,2): 198 (3,0): 3 (3,1): 206 (3,2): 457 (3,3): 50 (3,4): 108

witness diameter X 4.0 · Z 3.1623 (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

layout contributed by @vprusso · Inherited from the laid-out parent codes/126-6-5.json: the parent's coordinates with the entry of the removed qubit (parent index 0) dropped, since this peel child keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04

X checkZ checkqubit site (125)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 priced calculus E-PEEL transform of codes/126-6-5.json

model Beam 1.0 (claimed, not verified)

date 2026-11-02

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

[[125,6,4]] — priced calculus E-PEEL transform of codes/126-6-5.json

Direction & hypothesis

Weight-4 unrestricted CSS cell. Transform a connected CSS parent (codes/126-6-5.json, a board entry) with the priced construction calculus to obtain a connected child that is not dominated by any existing entry in its track cell.

What was searched

Gap-targeted search: all priced calculus moves (peel, puncture, deletion, rebase) applied to board codes. Targets Pareto frontier gaps. Rule used: E-PEEL. Parent: codes/126-6-5.json. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side, then verify/validate_candidate.py with full refutation.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. Parent entry: codes/126-6-5.json.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer.

Reproduction

Rebuild H_X/H_Z from codes/126-6-5.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 59 (max weight 4) · Z-checks 60 (max weight 4)

H_X (59 checks, sparse supports)

[1, 2] [4, 5] [6, 7] [0, 1, 10, 11] [2, 3, 12, 13] [5, 6, 15, 16] [7, 8, 17, 18] [9, 10, 19, 20] [11, 12, 21, 22] [14, 15, 24, 25] [16, 17, 26, 27] [20, 21, 30, 31] [22, 23, 32, 33] [25, 26, 36, 37] [27, 28, 38, 39] [29, 30, 40, 41] [31, 32, 42, 43] [33, 34, 44, 45] [35, 36, 46, 47] [37, 38, 48, 49] [41, 42] [43, 44, 51, 52] [45, 46, 53, 54] [47, 48, 55] [49, 50] [56, 57] [51, 58, 59] [52, 53, 60, 61] [54, 55, 62, 63] [64, 65] [57, 58, 68, 69] [59, 60, 70, 71] [61, 62, 72, 73] [63, 64, 74, 75] [65, 66, 76, 77] [67, 68, 78, 79] [69, 70, 80, 81] [73, 74, 83, 84] [75, 76, 85, 86] [79, 80, 89, 90] [81, 82, 91, 92] [84, 85, 95, 96] [86, 87, 97, 98] [88, 89, 99, 100] [90, 91, 101, 102] [92, 93, 103, 104] [94, 95, 105, 106] [96, 97, 107, 108] [100, 101] [102, 103, 110, 111] [104, 105, 112, 113] [106, 107, 114] [108, 109] [110, 115] [111, 112, 116, 117] [113, 114, 118, 119] [115, 116, 120, 121] [117, 118, 122, 123] [119, 124]

H_Z (60 checks, sparse supports)

[0, 9, 10] [1, 2, 11, 12] [3, 13] [4, 5, 14, 15] [6, 7, 16, 17] [8, 18] [9, 19] [10, 11, 20, 21] [12, 13, 22, 23] [14, 24] [15, 16, 25, 26] [17, 18, 27, 28] [19, 20, 29, 30] [21, 22, 31, 32] [23, 33, 34] [24, 25, 35, 36] [26, 27, 37, 38] [28, 39] [29, 40] [30, 31, 41, 42] [32, 33, 43, 44] [34, 35, 45, 46] [36, 37, 47, 48] [38, 39, 49, 50] [44, 45, 52, 53] [46, 47, 54, 55] [51, 52, 59, 60] [53, 54, 61, 62] [56, 57, 67, 68] [58, 59, 69, 70] [60, 61, 71, 72] [62, 63, 73, 74] [64, 65, 75, 76] [66, 77] [67, 78] [68, 69, 79, 80] [70, 71, 81, 82] [72, 73, 83] [74, 75, 84, 85] [76, 77, 86, 87] [78, 79, 88, 89] [80, 81, 90, 91] [82, 92, 93] [83, 84, 94, 95] [85, 86, 96, 97] [87, 98] [88, 99] [89, 90, 100, 101] [91, 92, 102, 103] [93, 94, 104, 105] [95, 96, 106, 107] [97, 98, 108, 109] [103, 104, 111, 112] [105, 106, 113, 114] [110, 111, 115, 116] [112, 113, 117, 118] [116, 117, 121, 122] [118, 119, 123, 124] [120, 121] [122, 123]

Code ID 125-6-4 · download JSON · raw on GitHub