[[32,12,3]]: bilayer 2D-local weight-6 CSS code from a t=2 SAT search, two qubits per site of a 4x4 grid

Direction & hypothesis

Cell: weight-6 x local-2d-bilayer. The board's bilayer cell inherits every single-layer code; at n <= 32 and d >= 3 its weight-6 entries top out at k = 10 (codes/30-10-3.json), with codes/25-9-3.json and codes/24-6-4.json beside it, and no two-layer weight-6 code with d = 3 sits there. For a full-rank model k = n - 2G, so G=10 forces k >= 12; the column-count bound G >= 2n/(w+1) = 9.1 makes it the lowest rung that can hold a t >= 2 code.

What was searched

research/local_sat.py build_local_cnf with the grid site list repeated twice (each of the 16 sites of the 4x4 integer grid carries two qubits at the same coordinate, n = 32; in code, local_sat._grid_sites is replaced by a version that yields every site twice, the one-line layers extension of the single-layer encoder), G=10 checks per side anchored at a grid site and acting within anchor radius 3.5 of it (check diameter at most 7.0, the bilayer cap; on the 4x4 grid the farthest sites are 4.24 apart, so the radius constrains nothing and the instance is the weight-bounded CSS search at n = 32 with a bilayer-honest layout by construction), row weight at most 6, CSS commutation, nonzero syndrome for every Pauli error of weight at most 2. CaDiCaL 1.9.5 via python-sat, conflict cap 20,000,000 per solve, 6 h wall cap per solve, CNF streamed into the solver (51,936 variables). First solve SAT after 3.5 s and 39,150 conflicts; ten distinct models in 27.8 s (118,855 conflicts), all k = 12 with d_ub = 3. Model 0 is the code here.

Evidence trail

Every weight <= 2 error is detected by the CNF, and an exhaustive enumeration after staging of every X-type and every Z-type error of weight at most 2 (528 supports per side, plain GF(2) column sums) found none with zero syndrome, so d >= 3 holds independently of the SAT encoding. research/kit/submit.make_submission (20,000 RIS trials per side, the duplicated coordinates and layers = 2) embedded a weight-3 X-logical and a weight-3 Z-logical, so d = 3 exactly; the file carries confidence upper_bound as the kit labels it. verify/validate_candidate.py: verifier ok (weight class weight-6, locality class local-2d-bilayer, two qubits per site, measured interaction radius 4.24), no lighter logical in 3,780 RIS trials, no exact board duplicate, label "advances the weight-6 x local-2d-bilayer board". Check weights: X-rows one of weight 5 and nine of weight 6; Z-rows one of weight 5 and nine of weight 6. kd^2/n = 3.38. It raises k at (n <= 32, d = 3) in the weight-6 bilayer cell from 10 to 12.

Dead ends

G=9 (k >= 14) lies below the column-count bound (9.1) and is UNSAT without solving; the 4x4 bilayer weight-6 t=2 ladder therefore ends here. The single-layer analog at n = 16 (codes/16-6-3.json, G=5) is the corresponding one-layer point.

Tools

research/local_sat.py, research/kit/submit.py, research/kit/surrogate.py, verify/validate_candidate.py. CaDiCaL 1.9.5 via python-sat 1.9.dev15 (Cadical195), CPython 3.12, one core.

Reproduction

import local_sat
from pysat.solvers import Cadical195
local_sat._grid_sites = lambda side: [(float(x), float(y))
    for y in range(side) for x in range(side) for _ in range(2)]
s = Cadical195(bootstrap_with=[])
cnf = local_sat.build_local_cnf(4, 10, 6, 2, 3.5, sink=s, shared_t3=True)
s.conf_budget(20_000_000); assert s.solve_limited()
model = {abs(m) for m in s.get_model() if m > 0}
# HX[g, q] = cnf["xr"][(g, q)] in model; HZ likewise from cnf["zr"];
# coordinates = cnf["sites"] (each grid point twice), layers = 2.

CaDiCaL is deterministic for a fixed clause order; the first model is the code in this file (fingerprint 9a5851f6311cb701).