[[173,4,17]] · QEC Challenge

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[[173,4,17]] d ≤

173

kd²/n

6.68208

X/Z

1.06

0.004

6.4031

layers

swaps

997

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Distance

X/Z asymmetry 1.06 · d_X ≤ 18, d_Z ≤ 17 · 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 18 · witness weight 18 (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, 18 qubits)

[12, 13, 54, 63, 72, 82, 90, 95, 99, 110, 127, 142, 144, 147, 150, 162, 165, 171]

d_Z 17 · witness weight 17 (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, 17 qubits)

[2, 5, 19, 39, 74, 83, 85, 86, 89, 101, 102, 109, 111, 130, 140, 144, 148]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 6 · H_Z 5–6 (mean 5.976)

qubit degrees H_X 2–3 (mean 2.913) · H_Z 1–3 (mean 2.936)

trapping sets H_X (1,2)×15 (2,2)×3 (3,2)×18 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 15 (1,3): 158 (2,2): 3 (2,3): 144 (2,4): 1113 (3,2): 18 (3,3): 251 (3,4): 1623 (3,5): 9975 (3,6): 276 (3,7): 1392

trapping sets H_Z (1,1)×1 (2,2)×6 (3,2)×13 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 1 (1,2): 9 (1,3): 163 (2,2): 6 (2,3): 88 (2,4): 1171 (3,2): 13 (3,3): 251 (3,4): 1023 (3,5): 10699 (3,6): 172 (3,7): 1494

witness diameter X 10.2956 · Z 10.8167 (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/174-4-18.json: the parent's coordinates with the entry of the removed qubit (parent index 86) 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 (88)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 997 nearest-neighbor SWAPs per round in total, at most 12 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/174-4-18.json

model Beam 1.0 (claimed, not verified)

date 2026-11-02

family other (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

[[173,4,17]] — priced calculus E-PEEL transform of codes/174-4-18.json

Direction & hypothesis

Weight-6 unrestricted CSS cell. Transform a connected CSS parent (codes/174-4-18.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/174-4-18.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/174-4-18.json.

Tools

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

Reproduction

Rebuild H_X/H_Z from codes/174-4-18.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 84 (max weight 6) · Z-checks 85 (max weight 6)

H_X (84 checks, sparse supports)

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

H_Z (85 checks, sparse supports)

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

Code ID 173-4-17 · download JSON · raw on GitHub