[[108,12,6]] · QEC Challenge

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[[108,12,6]] d ≤stabilizer

108

kd²/n

4.0

0.0053

7.0711

layers

swaps

216

Share this result

Distance

a general stabilizer code has no X and Z sides: d is the minimum Pauli weight of a nontrivial logical operator (a Y factor counts one qubit), and the witness is one Pauli operator that commutes with every generator and is not a product of them

d 6 · witness Pauli weight 6 (claimed upper_bound)

witness operator (Pauli string, 6 qubits)

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certificate none yet · distance stands as a self-certified upper bound (d ≤); the Pauli-weight certifier is not built yet, so stabilizer entries cannot earn d= for now

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth S 4 (shortest cycle of the generator Tanner graph; longer is friendlier to belief propagation)

check weights S 6

qubit degrees S 6

trapping sets S (1,6)×108 (2,8)×648 (3,10)×4104 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,6): 108 (2,8): 648 (2,10): 324 (3,10): 4104 (3,12): 6588 (3,14): 1296

witness diameter P 5.099 (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 3D layout

as measured by the verifier over the submitted [x, y, z] coordinates; a 3D layout earns no 2D-local class and is priced by the D = 3 geometric efficiency g = 2√2kd/(nρr³)

interaction radius 7.0711

bounding box 5.0 × 5.0 × 5.0

min site spacing 1.4142

max qubits per site 1 (layers 1)

qubits per unit volume 0.864

Construction & provenance

authors Chamon, Claudio

provenance literature baseline

construction Chamon code (arXiv:cond-mat/0404182) on an L = 6 periodic cubic lattice: qubits on even-parity sites (n = L³/2), one weight-6 generator per odd site, X on its +-x neighbours, Y on +-y, Z on +-z

model Claude Fable 5.1 (claimed, not verified)

reference arXiv:cond-mat/0404182, arXiv:1006.4871

date 2005

notes Literature baseline reconstruction of Chamon's model as a stabilizer code, k = 2L = 12, d <= L = 6 (RIS Pauli-weight witness); Bravyi-Leemhuis-Terhal (arXiv:1006.4871) analyse the model. Genuinely non-CSS (every generator carries X, Y and Z); no layout filed because of the periodic wrap-around.

family topological (a tag, not a ranking)

locality unrestricted (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

[[108,12,6]] — Chamon code on an L = 6 periodic cubic lattice

Direction & hypothesis

A literature baseline for the general-stabilizer board. Chamon's model (arXiv:cond-mat/0404182) is the standard genuinely non-CSS topological stabilizer code: every generator carries X, Y and Z, and it is not a Hadamard image of a CSS code. Bravyi, Leemhuis and Terhal (arXiv:1006.4871) analysed it as a code (fracton-type: k grows with the linear size).

What was searched

Nothing searched: the code is a reconstruction. Qubits sit on the even-parity sites of an L x L x L cubic lattice with periodic boundaries (n = L^3 / 2); for each odd-parity site c there is one generator X on c +- e_x, Y on c +- e_y, Z on c +- e_z (weight 6). We computed k = n - rank of the symplectic matrix (12 = 2L here) and searched for the distance.

Evidence trail

qldpc submit's Pauli-weight RIS search (20,000 trials) found a logical of weight 6 = L;

the JSON witness has that weight. Upper bound: the board's certifier does not minimise Pauli weight. k d^2 / n = 4 for every L in this family (k = 2L, d <= L, n = L^3 / 2).

Same construction at L = 6 gives [[108,12,<=6]] and at L = 8 [[256,16,<=8]].

Dead ends

Not applicable (reconstruction). No 3D layout is filed: the coordinates are periodic, so a literal embedding gives an interaction radius of order L at the wrap-around; folding the 3-torus is left open.

Tools

Claude Fable 5.1 in Claude Code; numpy; this repository's cli/qldpc.py and verify/.

Reproduction

L = 6. even = {(x,y,z) : x+y+z even}, indexed in lexicographic order; for each odd site (x,y,z): X at (x+-1,y,z), Y at (x,y+-1,z), Z at (x,y,z+-1), all coordinates mod L. S = (A | B) with A the X/Y support and B the Z/Y support.

Stabilizer generators

generators 108 (max weight 6; 108 mixed X/Z) (one binary symplectic matrix S = (A | B); generator i is X on A_i and Z on B_i, Y where both)

generators (108, Pauli strings on 108 qubits)

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IIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIYIIZZIYIIIII IIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIYIIZZIYIIII IIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIYZIZIIYIII IIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIYIIZIZYII IIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIYIZZIIYI IIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIYIZZIIY IIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIIIYIIZZI IIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIIIYIIZZ IIIIIIIIIIIIIIIIIXIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIXIIYIIIIIIIIIIIYZIZ

symplectic rows (A | B) (108, sparse supports)

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

Code ID 108-12-6 · download JSON · raw on GitHub