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[[101,5,5]] d =
n
101
k
5
d
5
kd²/n
1.238
w
4
X/Z
1
g
1.24
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · 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 operator (support, 5 qubits)
[14, 29, 46, 63, 79]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[41, 59, 77, 90, 99]
certificate exact, d = 5 · CryptoMiniSat 5.14.7 SAT
X: no logical < 5 exists; Z: no logical < 5 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 2–4 (mean 3.542) · H_Z 2–4 (mean 3.542)
qubit degrees H_X 1–2 (mean 1.683) · H_Z 1–2 (mean 1.683)
trapping sets H_X (1,1)×32 (2,0)×10 (3,0)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 32 (1,2): 69 (2,0): 10 (2,1): 64 (2,2): 158 (3,0): 2 (3,1): 168 (3,2): 369 (3,3): 38 (3,4): 88
trapping sets H_Z (1,1)×32 (2,0)×10 (3,0)×2 (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): 69 (2,0): 10 (2,1): 64 (2,2): 158 (3,0): 2 (3,1): 168 (3,2): 369 (3,3): 38 (3,4): 88
witness diameter X 4.1231 · Z 4.1231 (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)

Circuit tier

syndrome-extraction memory circuits committed under circuits/101-5-5/ · canonical noise recipe, 5 rounds, stim 1.16.0
d_circ ≤ 5 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 5 · fault-set witness of 5 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 5)
[108, 110, 151, 185, 207]
d_circ^Z 5 · fault-set witness of 5 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 5)
[220, 221, 432, 685, 922]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 1.414
X checkZ checkqubit site (101)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 known parameter set; see provenance notes
construction Dense-packed surface code (arXiv:2511.06758): five rotated surface-code patches fused via code deformation into one weight-4 patch, reconstructed from the authors' stim simulation (github.com/kohei-fujiu/Dense_Pakcing). Data qubits at odd,odd sites of the five_dense_num lattice; each stabilizer on the four diagonal neighbors of its ancilla.
builds on arXiv:2511.06758
date 2026-08-13
notes Reconstruction of the dense-packed surface code of arXiv:2511.06758 (Fujiu et al.), from the authors' released stim simulation (github.com/kohei-fujiu/Dense_Pakcing). The [[n,k,d]] parameter set is the paper's construction; this entry adds an honest single-layer layout and verifier-checked witnesses. Not a new code family.
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

[[101,5,5]] — dense-packed surface code (5 logical qubits in one patch)

Direction & hypothesis

Target: the local-2d-single / weight-4 cell, the geometric-efficiency track. The board's weight-4 single-layer codes top out at `g = 4kd²/(n·ρ²·r⁴) ≈ 1.14` ([[672,85,3]]) and the surface-code baselines sit at exactly 1.0. Paper arXiv:2511.06758 (Fujiu et al., "Dense packing of the surface code") fuses five distance-d rotated surface-code patches into one contiguous patch via code deformation, keeping each logical's distance at d while sharing bulk stabilizer regions — the physical-qubit-per-logical overhead drops to ~3/4 of standalone patches. If the fused layout keeps the nearest-neighbour tilted lattice (r = √2, ρ = 1), the n-savings translate directly into a higher g than anything currently on the cell.

What was searched

  • Reconstructed the code faithfully from the authors' released simulation

    • (github.com/kohei-fujiu/Dense_Pakcing, dense_packing_simulation_x_error.py): five_dense_num site mask, data qubits at odd,odd sites, Z-ancillas at (x+y)%4==2 (measure X-checks), all other non-data sites are X-ancillas (measure Z-checks); each stabilizer acts on the four diagonal data neighbours of its ancilla.

  • Instances: d = 5 (n = 101) and d = 7 (n = 197), both k = 5, weight 4.
  • GF(2) sanity: CSS commutation exact, rank(H_X) = rank(H_Z) = 48 (d=5), so

    • k = 101 − 48 − 48 = 5, matching the five codewords the paper claims.

  • Layout: data qubit q_ij at grid site (x,y); nearest-neighbour diagonal

    • checks span √2. Honest single layer, 1 qubit/site, no cramming.

Evidence trail

Distance confirmation ladder (d = 5 instance, 101 data qubits):

  • Exact side certification (this work): exhaustive enumeration of all weight

    • ≤ 4 vectors commuting with the opposite stabilizer group found no X- or Z-logical of weight < 5 (weight 1: 0, weight 2: 10, weight 3: 2, weight 4: 98 vectors commute with all Z-stabilizers, all in the stabilizer rowspace). The paper's own logical operators (its OBSERVABLE_INCLUDE lines, weight 5 each) are genuine logicals — in ker(H_other), outside rowspace(H_self). Hence d_X = d_Z = 5 exactly, no trust required.

  • CLI RIS gate (20000 trials) independently found d ≤ 5 on both sides,

    • consistent.

  • The d = 7 instance is witness-backed d ≤ 7 (not exhaustively certified

    • here; RIS 20000 trials agrees).

Claim: exact for [[101,5,5]] (both sides), upper_bound for [[197,5,7]].

What it advances

  • Geometric efficiency g on local-2d-single/weight-4: g = 1.238,

    • above the previous best on the cell ([[672,85,3]], g ≈ 1.14) and above the surface-code baselines (g = 1.00). The 5-logical pack needs n = 101 vs n = 125 for five standalone d-5 patches.

  • kd²/n = 1.238 — modest; this is a density win, not a rate win.
  • The construction family (topological, dense packing) is new on the board.

Dead ends

  • A naive "fused patch = surface code with shared region" reading of the

    • paper's Fig. 6 is wrong for scoring: the shared region is NOT a separate code, and the fused object has k = 5 (not 25). Early rank mistakes (counting ancillas as code qubits) gave k = 174 — the verifier's n = data-qubit rule is the correct frame: n = 101 data qubits, k = 5.

  • A guessed interaction radius r ≈ 2.8 (long junction reaches) is also wrong:

    • the actual construction stays on the tilted nearest-neighbour lattice, so r = √2 and the g gain survives the r⁴ penalty. Verified against the layout from the authors' own QUBIT_COORDS.

Tools

  • Reconstruction: faithful port of the authors' stim-circuit geometry into a

    • standalone GF(2) generator (see Reproduction).

  • numpy GF(2) rref for ranks; exhaustive weight ≤ 4 logical search

    • (itertools combinations over 101 qubits, ~4M checks per side) for the exact d = 5 certification.

  • Repo tooling: ./qldpc submit (RIS witness search, schema-valid JSON,

    • verifier, locality derivation).

Reproduction

python research/build_dense_surface.py 5   # writes /tmp/dense_5.npz (hx, hz, coords)
./qldpc submit /tmp/dense_5.npz --coords /tmp/dense_5.npz --layers 1 \
  --authors @mathysrennela --family topological

research/build_dense_surface.py in this repo reproduces the matrices exactly from dense_packing_simulation_x_error.py (five_dense_num, data_num, auxiliary_z masks and diagonal-neighbour supports). The d = 7 instance: python research/build_dense_surface.py 7.

Parity checks

X-checks 48 (max weight 4) · Z-checks 48 (max weight 4)
H_X (48 checks, sparse supports)
[0, 1] [2, 3] [5, 6] [7, 8] [10, 11] [12, 13] [1, 2, 16, 17] [3, 4, 18, 19] [6, 7, 21, 22] [8, 9, 23, 24] [11, 12, 26, 27] [13, 14, 28, 29] [15, 16, 30, 31] [17, 18, 32, 33] [20, 21, 36, 37] [22, 23, 38, 39] [25, 26, 42, 43] [27, 28, 44, 45] [31, 32, 48, 49] [33, 34, 50, 51] [35, 36, 52, 53] [37, 38, 54, 55] [39, 40, 56, 57] [41, 42, 58, 59] [43, 44, 60, 61] [45, 46, 62, 63] [47, 48, 64, 65] [49, 50, 66, 67] [51, 52, 68, 69] [53, 54, 70, 71] [55, 56, 72, 73] [57, 58, 74, 75] [59, 60, 76, 77] [61, 62, 78, 79] [65, 66] [67, 68, 81, 82] [69, 70, 83, 84] [71, 72, 85] [73, 74, 86, 87] [75, 76, 88, 89] [77, 78, 90] [79, 80] [81, 91] [82, 83, 92, 93] [84, 85, 94, 95] [86, 96] [87, 88, 97, 98] [89, 90, 99, 100]
H_Z (48 checks, sparse supports)
[0, 1, 15, 16] [2, 3, 17, 18] [4, 19] [5, 6, 20, 21] [7, 8, 22, 23] [9, 24] [10, 11, 25, 26] [12, 13, 27, 28] [14, 29] [15, 30] [16, 17, 31, 32] [18, 19, 33, 34] [20, 35, 36] [21, 22, 37, 38] [23, 24, 39, 40] [25, 41, 42] [26, 27, 43, 44] [28, 29, 45, 46] [30, 31, 47, 48] [32, 33, 49, 50] [34, 35, 51, 52] [36, 37, 53, 54] [38, 39, 55, 56] [40, 41, 57, 58] [42, 43, 59, 60] [44, 45, 61, 62] [46, 63] [47, 64] [48, 49, 65, 66] [50, 51, 67, 68] [52, 53, 69, 70] [54, 55, 71, 72] [56, 57, 73, 74] [58, 59, 75, 76] [60, 61, 77, 78] [62, 63, 79, 80] [68, 69, 82, 83] [70, 71, 84, 85] [74, 75, 87, 88] [76, 77, 89, 90] [81, 82, 91, 92] [83, 84, 93, 94] [86, 87, 96, 97] [88, 89, 98, 99] [92, 93] [94, 95] [97, 98] [99, 100]
Code ID 101-5-5 · download JSON · raw on GitHub