Logical qubits k, distance d, physical qubits n. It rewards encoding more logical qubits and tolerating more errors, while penalising qubit overhead — the single number that captures whether a code is a good deal. Baseline: IBM's Gross code scores 12. Select a row to inspect the code, its lattice, and how its distance was verified.
| Code [[n,k,d]] | k·d²/n | vs Gross | Distance | qubits/log. | Novelty |
|---|---|---|---|---|---|
| [[144, 12, 12]] · Gross | 12.0 | 1.00× | baseline | 12.0 | IBM 2024 |
| [[196, 18, 24]] | 52.9 | 4.41× | bound | 10.9 | novel |
| [[192, 16, 20]] | 33.3 | 2.78× | bound | 12.0 | novel |
| [[196, 8, 26]] | 27.6 | 2.30× | bound | 24.5 | novel |
| [[192, 8, 24]] | 24.0 | 2.00× | exact | 24.0 | novel |
| [[168, 8, 22]] | 23.1 | 1.92× | bound | 21.0 | cyclic risk |
| [[180, 6, 26]] | 22.5 | 1.88× | bound | 30.0 | cyclic risk |
| [[198, 6, 27]] | 22.1 | 1.84× | bound | 33.0 | cyclic risk |
Seven codes from a 60,000-candidate search; 14 of the top 50 beat Gross after rigorous re-verification. Distances are exact only where marked — fast decoders upper-bound the distance and can over-state it.
Distance — MIP exact — every one of 255 cosets solved (CBC). BP+OSD had over-estimated d=26; the solver corrected it to 24.
The certified result. Asymmetric distance (d_X=30, d_Z=24) — valuable under biased noise. The only code here whose distance is proven exactly.
A higher figure of merit buys less margin elsewhere. Our top codes win decisively on k·d²/n — and currently lose on the circuit-level threshold, which sits below today's hardware error rates. Closing that gap (a fault-tolerant schedule, full-scale runs) is the next phase. The tradeoff is the finding.
Threshold figures are preliminary (naive, non-fault-tolerant syndrome schedule); exact values await the next phase.
Ranked #1 on the raw leaderboard at a claimed distance of 20 (score 41.67). Rigorous BP+OSD verification dropped it to distance 8 (score 6.67). We publish failures like this — an unverified figure of merit is worth nothing.
Faithful to the program's open release (code · data · demos). Every figure here is reproducible; distances are labelled exact (MIP-certified) or bound (BP+OSD upper bound).