[[108,8,10]] · QEC Challenge

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[[108,8,10]] d =

108

kd²/n

7.407

X/Z

0.0057

6.0

layers

swaps

535

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Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · 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 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[3, 26, 41, 42, 61, 68, 82, 83, 98, 106]

d_Z 10 · witness weight 10 (claimed upper_bound)

witness operator (support, 10 qubits)

[3, 10, 19, 27, 42, 43, 68, 83, 84, 99]

certificate exact, d = 10 · scipy/HiGHS MILP

X: no logical < 10 exists; Z: no logical < 10 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 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 6

qubit degrees H_X 3 · H_Z 3

trapping sets H_X (1,3)×108 (2,4)×810 (3,3)×126 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 108 (2,4): 810 (3,3): 126 (3,5): 7722 (3,7): 1080

trapping sets H_Z (1,3)×108 (2,4)×810 (3,3)×126 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 108 (2,4): 810 (3,3): 126 (3,5): 7722 (3,7): 1080

witness diameter X 4.1231 · Z 5.831 (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 (54)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 535 nearest-neighbor SWAPs per round in total, at most 9 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 Bravyi, Sergey and Cross, Andrew W. and Gambetta, Jay M. and Maslov, Dmitri and Rall, Patrick and Yoder, Theodore J.

provenance literature baseline

construction Bivariate bicycle code, orders (9, 6), A=x³ + y² + y, B=x² + x + y³ (periodic boundary conditions).

model classical construction (no AI model)

reference arXiv:2308.07915

date 2023-08-15

notes Distance d=10 established in arXiv:2308.07915 (Table 3); witness here is a decoder-found logical operator of that weight (upper bound).

family bivariate bicycle (a tag, not a ranking)

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

weight class weight ≤ 6 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 54 (max weight 6) · Z-checks 54 (max weight 6)

H_X (54 checks, sparse supports)

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

H_Z (54 checks, sparse supports)

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

Code ID 108-8-10 · download JSON · raw on GitHub