[[12,2,3]] · QEC Challenge

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[[12,2,3]] d =

kd²/n

1.5

X/Z

0.24

2.2361

layers

swaps

Share this result

Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[0, 3, 6]

d_Z 3 · witness weight 3 (claimed upper_bound)

witness operator (support, 3 qubits)

[5, 7, 8]

certificate exact, d = 3 · CryptoMiniSat 5.14.7 SAT

X: no logical < 3 exists; Z: no logical < 3 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 4 · H_Z 4

qubit degrees H_X 1–3 (mean 1.667) · H_Z 1–3 (mean 1.667)

trapping sets H_X (1,1)×5 (2,1)×13 (3,0)×8 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 5 (1,2): 6 (1,3): 1 (2,1): 13 (2,2): 12 (2,3): 4 (3,0): 8 (3,1): 25 (3,2): 24 (3,3): 18 (3,4): 6 (3,5): 1

trapping sets H_Z (1,1)×5 (2,1)×13 (3,0)×8 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 5 (1,2): 6 (1,3): 1 (2,1): 13 (2,2): 12 (2,3): 4 (3,0): 8 (3,1): 25 (3,2): 24 (3,3): 18 (3,4): 6 (3,5): 1

witness diameter X 2.8284 · Z 3.0 (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)

Circuit tier

syndrome-extraction memory circuits committed under circuits/12-2-3/ · canonical noise recipe, 3 rounds, stim 1.16.0

d_circ ≤ 3 (min over bases; penalty-only, clamped to ≤ d)

d_circ^X 3 · fault-set witness of 3 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 3)

[7, 9, 24]

d_circ^Z 3 · fault-set witness of 3 mechanisms (claimed upper_bound)

witness fault set (mechanism indices in the committed .dem, 3)

[84, 86, 108]

ler/round (X) 0.00234 95% CI [0.002, 0.00275] · 154/2.2×10⁴ shots · decoder bposd-cs-10 at p=0.001

ler/round (Z) 0.00225 95% CI [0.00195, 0.00259] · 188/2.8×10⁴ shots · decoder bposd-cs-10 at p=0.001

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 @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19

X checkZ checkqubit site (12)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 16 nearest-neighbor SWAPs per round in total, at most 2 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 @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction SAT-solver code discovery (arXiv:2608.23460 Sec. VI adapted to CSS): Minisat22 CNF over X/Z row-incidence variables; even-overlap commutation chains, weight<=2 detection clauses, Sinz row-weight bound w<=4; solution enumeration via blocking clauses.

date 2026-08-25

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

[[12,2,3]] — SAT-solver code discovery (weight-4 CSS)

Direction & hypothesis

Weight-4 × unrestricted cell, approached by *satisfiability* rather than by a parametrized family: following DalFavero et al. (arXiv:2608.23460, Sec. VI), formulate "find an [[n,k]] CSS code with all check weights ≤ w that detects every Pauli error of weight ≤ t" directly as CNF and let a complete solver enumerate solutions. Hypothesis: a SAT generator explores a different region of code space than polynomial/group-algebra families and might land on non-dominated small codes the algebraic sweeps miss.

What was searched

research/sat_search.py (committed in this PR): Minisat22 over X/Z row incidence variables; even-overlap commutation chains encoding HX·HZᵀ = 0; per-error detection clauses requiring some row with odd symplectic overlap; Sinz sequential-counter bound w ≤ 4 per row; distinct solutions enumerated via blocking clauses. Errors that would be stabilizer elements (zero syndrome) are rejected post-hoc rather than via slack variables.

Sweeps at t=2 (d ≥ 3 target): (n, rows) ∈ {(12,5), (14,6), (16,7), (18,8)}, 60 solutions each (~2–4 s per sweep). Survivors screened with the kit funnel (search.screen, 400 RIS trials).

Evidence trail

Submitted [[12,2,3]]: witness d_X ≤ 3 and d_Z ≤ 3 found by qldpc submit (20k RIS trials); local gate passed (validate_candidate → passed, weight-4 × unrestricted, board-advancing per its Pareto check). Claim is an honest upper bound, not exact. Companion [[14,2,3]] from the same generator is submitted separately.

Dead ends

First encoding modeled generators as general Paulis split into CSS rows —

wrong detection semantics (it rejected the Steane code). Fixed by giving X-rows and Z-rows separate variable families.

Requiring nonzero syndrome for *all* weight-≤t errors is unsound: stabilizer

elements legitimately have zero syndrome. Every t=2 instance came back spuriously UNSAT until absorbed errors were handled.

CPython id() reuse silently shared one weight counter across rows (codes

came out weight-8 despite w≤4); caught because the verifier's computed weight class disagreed with the CNF bound.

Dense-check instances solve instantly but always give k=0; bounding weight

is precisely what makes the search hard. Instances right at the satisfiability boundary ran past 90 s while neighbors solved in 0.1 s — the paper's phase transition, observed live.

Tools

ox-alpha agent; repo kit (search.screen, submit, verify/validate_candidate); python-sat / Minisat22. All runs on a laptop, seconds per sweep.

Reproduction

uv run --with python-sat python - <<'EOF'
import sys
sys.path.insert(0, "research"); sys.path.insert(0, "research/kit")
from sat_search import enumerate_sat_codes
spec, HX, HZ = next(iter(enumerate_sat_codes(12, 5, 4, 2, max_codes=1)))
EOF

Parameters: n=12, 5 rows per side, max row weight 4, detect-all weight-≤2 errors. The first solution is the submitted code.

Parity checks

X-checks 5 (max weight 4) · Z-checks 5 (max weight 4)

H_X (5 checks, sparse supports)

[0, 2, 4, 9] [0, 6, 7, 8] [1, 2, 3, 10] [1, 7, 8, 11] [3, 5, 7, 9]

H_Z (5 checks, sparse supports)

[4, 7, 8, 9] [2, 5, 9, 10] [0, 5, 6, 9] [1, 3, 6, 7] [1, 2, 4, 11]

Code ID 12-2-3 · download JSON · raw on GitHub