[[70,6,8]] · QEC Challenge

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[[70,6,8]] d =

kd²/n

5.486

X/Z

0.0027

6.7082

layers

swaps

202

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Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[5, 17, 39, 42, 46, 47, 52, 62]

d_Z 8 · witness weight 8 (claimed upper_bound)

witness operator (support, 8 qubits)

[0, 27, 29, 35, 51, 60, 62, 63]

certificate exact, d = 8 · scipy/HiGHS MILP

X: no logical < 8 exists; Z: no logical < 8 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 4 · H_Z 4 (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)×70 (2,2)×35 (3,3)×630 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 70 (2,2): 35 (2,4): 455 (3,3): 630 (3,5): 3360 (3,7): 420

trapping sets H_Z (1,3)×70 (2,2)×35 (3,3)×630 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 70 (2,2): 35 (2,4): 455 (3,3): 630 (3,5): 3360 (3,7): 420

witness diameter X 5.3852 · Z 6.4031 (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 (35)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 202 nearest-neighbor SWAPs per round in total, at most 10 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 Zijian Liang and Ke Liu and Hao Song and Yu-An Chen

provenance literature baseline

construction Twisted-torus bivariate-bicycle code from Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025). Stabilizers f(x,y)=1+x+xy, g(x,y)=1+y+xy^-1 on the abelian quotient group Z²/L with twist basis a_1=[0, 7], a_2=[5, 1] (n=2|det[a_1,a_2]|=2*35). Reconstructed from the published polynomials and twist; CSS form H_X=[f|g], H_Z=[gbar|fbar].

model classical construction (no AI model)

reference arXiv:2503.03827

date 2025

notes Reconstructed baseline from Liang, Liu, Song, Chen, Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025), arXiv:2503.03827. The [[n,k,d]] parameter set and stabilizer polynomials are published in that paper; this entry reproduces them. Distance is an upper bound per the paper (probabilistic for d>20); the witness is the surrogate logical of that weight. Seeded to fill a board coverage gap at this block size, not a discovery. Layout certification: a verifier-accepted honest bilayer layout (stacked 5x7 grid, 2 qubits per site, layers=2, measured interaction radius ~6.708) was found by @mathysrennela (Aug 2026). It earns the code the local-2d-bilayer class and makes the site's geometric efficiency g = 4kd^2/(n rho^2 r^4) computable. Code authorship unchanged; layout credit to @mathysrennela.

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 35 (max weight 6) · Z-checks 35 (max weight 6)

H_X (35 checks, sparse supports)

[0, 1, 4, 35, 37, 56] [1, 3, 7, 36, 39, 61] [2, 4, 8, 36, 37, 40] [3, 6, 11, 38, 42, 65] [4, 7, 12, 38, 39, 43] [5, 8, 13, 39, 40, 44] [6, 10, 16, 41, 46, 68] [7, 11, 17, 41, 42, 47] [8, 12, 18, 42, 43, 48] [9, 13, 19, 43, 44, 49] [0, 10, 15, 45, 51, 55] [11, 16, 22, 45, 46, 52] [12, 17, 23, 46, 47, 53] [13, 18, 24, 47, 48, 54] [14, 19, 25, 48, 49, 55] [1, 15, 21, 35, 50, 60] [0, 2, 16, 50, 51, 57] [17, 22, 27, 51, 52, 58] [18, 23, 28, 52, 53, 59] [19, 24, 29, 53, 54, 60] [20, 21, 25, 50, 54, 55] [3, 21, 26, 36, 56, 64] [2, 5, 22, 35, 57, 62] [23, 27, 31, 57, 58, 63] [24, 28, 32, 58, 59, 64] [25, 26, 29, 56, 59, 60] [6, 26, 30, 38, 61, 67] [5, 9, 27, 37, 62, 66] [28, 31, 34, 62, 63, 67] [29, 30, 32, 61, 63, 64] [10, 30, 33, 41, 65, 69] [9, 14, 31, 40, 66, 69] [32, 33, 34, 65, 66, 67] [15, 20, 33, 45, 49, 68] [14, 20, 34, 44, 68, 69]

H_Z (35 checks, sparse supports)

[0, 15, 22, 35, 45, 51] [1, 2, 21, 35, 36, 50] [0, 2, 27, 37, 51, 57] [3, 4, 26, 36, 38, 56] [1, 4, 5, 35, 37, 39] [2, 5, 31, 40, 57, 62] [6, 7, 30, 38, 41, 61] [3, 7, 8, 36, 39, 42] [4, 8, 9, 37, 40, 43] [5, 9, 34, 44, 62, 66] [10, 11, 33, 41, 45, 65] [6, 11, 12, 38, 42, 46] [7, 12, 13, 39, 43, 47] [8, 13, 14, 40, 44, 48] [9, 14, 33, 49, 66, 69] [15, 16, 20, 45, 50, 68] [10, 16, 17, 41, 46, 51] [11, 17, 18, 42, 47, 52] [12, 18, 19, 43, 48, 53] [13, 19, 20, 44, 49, 54] [10, 14, 20, 55, 68, 69] [0, 21, 25, 50, 55, 56] [16, 22, 23, 46, 52, 57] [17, 23, 24, 47, 53, 58] [18, 24, 25, 48, 54, 59] [15, 19, 25, 49, 55, 60] [1, 26, 29, 56, 60, 61] [22, 27, 28, 52, 58, 62] [23, 28, 29, 53, 59, 63] [21, 24, 29, 54, 60, 64] [3, 30, 32, 61, 64, 65] [27, 31, 32, 58, 63, 66] [26, 28, 32, 59, 64, 67] [6, 33, 34, 65, 67, 68] [30, 31, 34, 63, 67, 69]

Code ID 70-6-8 · download JSON · raw on GitHub