[[8,2,3]] · QEC Challenge

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[[8,2,3]] d ≤stabilizer

kd²/n

2.25

0.36

2.2361

layers

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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 3 · witness Pauli weight 3 (claimed upper_bound)

witness operator (Pauli string, 3 qubits)

ZZXIIIII X: [2] Z: [0, 1]

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 4

qubit degrees S 3

trapping sets S (1,3)×8 (2,2)×9 (3,1)×12 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 8 (2,2): 9 (2,4): 18 (3,1): 12 (3,3): 32 (3,5): 12

witness diameter P 1.4142 (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 @MicheldeMessieres · Simulated annealing over distinct integer-grid sites (one qubit per site) minimising the largest check diameter. Search driven by Claude Opus 5.5 (Claude Code). · 2026-10-09

stabilizer generator (qubit support)qubit site (8)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 3 nearest-neighbor SWAPs per round in total, at most 1 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 Cross, Andrew and Vandeth, Drew

provenance literature baseline

construction [[8,2,3]] weight-4 stabilizer code from Cross and Vandeth, 'Small Binary Stabilizer Subsystem Codes' (arXiv:2501.17447), as listed in Wei, Han, He, Li and Liu, 'Theory of low-weight quantum codes' (arXiv:2601.19848), Table 1. Generators as transcribed from that table.

model Claude Opus 5.5 (Claude Code) (claimed, not verified)

reference arXiv:2501.17447, arXiv:2601.19848

date 2026-10-09

notes Literature entry: the code is Cross and Vandeth's. Entered, with a one-layer 2D layout, by @MicheldeMessieres; the layout is credited in locality.contributed_by.

family other (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

[[8,2,3]] — weight-4 stabilizer code of Cross and Vandeth (literature entry)

Direction & hypothesis

Literature entry. The board's mixed-alarm (stabilizer) weight-4 cells had no copy of this published code, whose kd^2/n = 2.25 is above the cell's current best. The code is Cross and Vandeth's, from "Small Binary Stabilizer Subsystem Codes" (arXiv:2501.17447); we found it in Table 1 of Wei, Han, He, Li and Liu, "Theory of low-weight quantum codes" (arXiv:2601.19848), and transcribed its six weight-4 generators from that table.

What was searched

Nothing new was constructed.

Also builds on: the code is Cross and Vandeth's (arXiv:2501.17447), found in Table 1 of Wei, Han, He, Li and Liu (arXiv:2601.19848). Our contribution: entering this published code and giving it a one-layer 2D layout (simulated annealing over distinct integer-grid sites, one qubit per site), credited in locality.contributed_by.

Evidence trail

Distance: an exhaustive check over all Pauli operators of weight 1 and 2 finds no nontrivial logical, and a weight-3 logical exists (the stored witness), so d = 3 exactly for these generators. k = 2 from the generator rank. The board verifier (verify/qldpc_verify.py) accepts the entry with locality class local-2d-single.

| effort (CPU-hours) | lightest logical found | |---|---| | < 0.001 (exhaustive over all weight <= 2 Paulis, then weight 3) | 3 (no logical of weight 1 or 2) |

Total distance work: under 0.001 CPU-hours; the exhaustive check needs no trial counts.

Dead ends

None for this entry.

Tools

Claude Opus 5.5 (Claude Code) working for @MicheldeMessieres; Python for the exhaustive distance check and the layout.

Reproduction

The six generators are listed in the entry's checks; they are the Table 1 generators of arXiv:2601.19848 for this code.

Stabilizer generators

generators 6 (max weight 4; 6 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 (6, Pauli strings on 8 qubits)

ZIIIIYZY IIIZXIXZ ZZIYYIII IZXIIXIZ IXZIZIZI XIZZIZII

symplectic rows (A | B) (6, sparse supports)

X: [5, 7] Z: [0, 5, 6, 7] X: [4, 6] Z: [3, 7] X: [3, 4] Z: [0, 1, 3, 4] X: [2, 5] Z: [1, 7] X: [1] Z: [2, 4, 6] X: [0] Z: [2, 3, 5]

Code ID 8-2-3 · download JSON · raw on GitHub