verified 27 stabilizer terms reproduce |T3>^{ot 6}, checked symbolically
Implied per-copy exponent γ ≤ 0.5000 against a published 0.5000, no improvement.
1 run; 0.001 CPU-hours; 0.001 h wall-clock; on Apple silicon laptop, one core; direct construction, no search.
Tensor product of the exact decompositions research/constructions/data/T3_m2_rank3.json (decomposition 0 of 1, coefficients fitted exactly by fit_coeffs.fit) (taken 3 times): χ(|T3⟩⊗2) ≤ 3 x χ(|T3⟩⊗2) ≤ 3 x χ(|T3⟩⊗2) ≤ 3 gives χ(|T3⟩⊗6) ≤ 27, exponent 0.5000. First witness for this cell; it is the sub-multiplicative value the board already implied and does not move the exponent. Terms are the exact products (x0 and l concatenated, W and Q block diagonal) and coefficients the exact products, built by research/constructions/product_witness.py.
this repository · stabrank research/constructions/product_witness.py · structured construction
{
"schema_version": "0.1",
"orbit": "T3",
"m": 6,
"direction": "upper",
"rank": 27,
"provenance": {
"author": "this repository",
"reference": "stabrank research/constructions/product_witness.py",
"method": "structured construction",
"date": "2026-09-23",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 0.001,
"wall_clock_hours": 0.001,
"runs": 1,
"hardware": "Apple silicon laptop, one core; direct construction, no search"
}
},
"notes": "Tensor product of the exact decompositions research/constructions/data/T3_m2_rank3.json (decomposition 0 of 1, coefficients fitted exactly by fit_coeffs.fit) (taken 3 times): chi(|T3>^2) <= 3 x chi(|T3>^2) <= 3 x chi(|T3>^2) <= 3 gives chi(|T3>^6) <= 27, exponent 0.5000. First witness for this cell; it is the sub-multiplicative value the board already implied and does not move the exponent. Terms are the exact products (x0 and l concatenated, W and Q block diagonal) and coefficients the exact products, built by research/constructions/product_witness.py.",
"witness": {
"terms": [
{
"k": 3,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
1,
0,
0
],
[
0,
1,
0
],
[
0,
0,
1
]
],
"l": [
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
0,
0,
0,
0,
1
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
1,
0,
0
],
[
0,
1,
0
],
[
0,
0,
2
]
],
"l": [
0,
0,
1
]
},
{
"k": 3,
"x0": [
0,
0,
0,
0,
0,
2
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
1,
0,
0
],
[
0,
1,
0
],
[
0,
0,
0
]
],
"l": [
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
0,
0,
1,
0,
0
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
1,
0,
0
],
[
0,
2,
0
],
[
0,
0,
1
]
],
"l": [
0,
1,
0
]
},
{
"k": 3,
"x0": [
0,
0,
0,
1,
0,
1
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
1,
0,
0
],
[
0,
2,
0
],
[
0,
0,
2
]
],
"l": [
0,
1,
1
]
},
{
"k": 3,
"x0": [
0,
0,
0,
1,
0,
2
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
1,
0,
0
],
[
0,
2,
0
],
[
0,
0,
0
]
],
"l": [
0,
1,
0
]
},
{
"k": 3,
"x0": [
0,
0,
0,
2,
0,
0
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
1,
0,
0
],
[
0,
0,
0
],
[
0,
0,
1
]
],
"l": [
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
0,
0,
2,
0,
1
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
1,
0,
0
],
[
0,
0,
0
],
[
0,
0,
2
]
],
"l": [
0,
0,
1
]
},
{
"k": 3,
"x0": [
0,
0,
0,
2,
0,
2
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
1,
0,
0
],
[
0,
0,
0
],
[
0,
0,
0
]
],
"l": [
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
1,
0,
0,
0,
0
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
2,
0,
0
],
[
0,
1,
0
],
[
0,
0,
1
]
],
"l": [
1,
0,
0
]
},
{
"k": 3,
"x0": [
0,
1,
0,
0,
0,
1
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
2,
0,
0
],
[
0,
1,
0
],
[
0,
0,
2
]
],
"l": [
1,
0,
1
]
},
{
"k": 3,
"x0": [
0,
1,
0,
0,
0,
2
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
2,
0,
0
],
[
0,
1,
0
],
[
0,
0,
0
]
],
"l": [
1,
0,
0
]
},
{
"k": 3,
"x0": [
0,
1,
0,
1,
0,
0
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
2,
0,
0
],
[
0,
2,
0
],
[
0,
0,
1
]
],
"l": [
1,
1,
0
]
},
{
"k": 3,
"x0": [
0,
1,
0,
1,
0,
1
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
2,
0,
0
],
[
0,
2,
0
],
[
0,
0,
2
]
],
"l": [
1,
1,
1
]
},
{
"k": 3,
"x0": [
0,
1,
0,
1,
0,
2
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
2,
0,
0
],
[
0,
2,
0
],
[
0,
0,
0
]
],
"l": [
1,
1,
0
]
},
{
"k": 3,
"x0": [
0,
1,
0,
2,
0,
0
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
2,
0,
0
],
[
0,
0,
0
],
[
0,
0,
1
]
],
"l": [
1,
0,
0
]
},
{
"k": 3,
"x0": [
0,
1,
0,
2,
0,
1
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
2,
0,
0
],
[
0,
0,
0
],
[
0,
0,
2
]
],
"l": [
1,
0,
1
]
},
{
"k": 3,
"x0": [
0,
1,
0,
2,
0,
2
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
2,
0,
0
],
[
0,
0,
0
],
[
0,
0,
0
]
],
"l": [
1,
0,
0
]
},
{
"k": 3,
"x0": [
0,
2,
0,
0,
0,
0
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
0,
0,
0
],
[
0,
1,
0
],
[
0,
0,
1
]
],
"l": [
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
2,
0,
0,
0,
1
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
0,
0,
0
],
[
0,
1,
0
],
[
0,
0,
2
]
],
"l": [
0,
0,
1
]
},
{
"k": 3,
"x0": [
0,
2,
0,
0,
0,
2
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
0,
0,
0
],
[
0,
1,
0
],
[
0,
0,
0
]
],
"l": [
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
2,
0,
1,
0,
0
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
0,
0,
0
],
[
0,
2,
0
],
[
0,
0,
1
]
],
"l": [
0,
1,
0
]
},
{
"k": 3,
"x0": [
0,
2,
0,
1,
0,
1
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
0,
0,
0
],
[
0,
2,
0
],
[
0,
0,
2
]
],
"l": [
0,
1,
1
]
},
{
"k": 3,
"x0": [
0,
2,
0,
1,
0,
2
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
0,
0,
0
],
[
0,
2,
0
],
[
0,
0,
0
]
],
"l": [
0,
1,
0
]
},
{
"k": 3,
"x0": [
0,
2,
0,
2,
0,
0
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
0,
0,
0
],
[
0,
0,
0
],
[
0,
0,
1
]
],
"l": [
0,
0,
0
]
},
{
"k": 3,
"x0": [
0,
2,
0,
2,
0,
1
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
0,
0,
0
],
[
0,
0,
0
],
[
0,
0,
2
]
],
"l": [
0,
0,
1
]
},
{
"k": 3,
"x0": [
0,
2,
0,
2,
0,
2
],
"W": [
[
1,
2,
0,
0,
0,
0
],
[
0,
0,
1,
2,
0,
0
],
[
0,
0,
0,
0,
1,
2
]
],
"Q": [
[
0,
0,
0
],
[
0,
0,
0
],
[
0,
0,
0
]
],
"l": [
0,
0,
0
]
}
],
"coeffs": [
"sqrt(3)/9",
"sqrt(3)*exp(2*I*pi/9)/9",
"sqrt(3)*exp(4*I*pi/9)/9",
"sqrt(3)*exp(2*I*pi/9)/9",
"sqrt(3)*exp(4*I*pi/9)/9",
"-sqrt(3)/18 + I/6",
"sqrt(3)*exp(4*I*pi/9)/9",
"-sqrt(3)/18 + I/6",
"sqrt(3)*exp(8*I*pi/9)/9",
"sqrt(3)*exp(2*I*pi/9)/9",
"sqrt(3)*exp(4*I*pi/9)/9",
"-sqrt(3)/18 + I/6",
"sqrt(3)*exp(4*I*pi/9)/9",
"-sqrt(3)/18 + I/6",
"sqrt(3)*exp(8*I*pi/9)/9",
"-sqrt(3)/18 + I/6",
"(-sqrt(3)*exp(2*I*pi/9) + 3*I*exp(2*I*pi/9))/18",
"(-sqrt(3)*exp(4*I*pi/9) + 3*I*exp(4*I*pi/9))/18",
"sqrt(3)*exp(4*I*pi/9)/9",
"-sqrt(3)/18 + I/6",
"sqrt(3)*exp(8*I*pi/9)/9",
"-sqrt(3)/18 + I/6",
"(-sqrt(3)*exp(2*I*pi/9) + 3*I*exp(2*I*pi/9))/18",
"(-sqrt(3)*exp(4*I*pi/9) + 3*I*exp(4*I*pi/9))/18",
"sqrt(3)*exp(8*I*pi/9)/9",
"sqrt(3)*exp(-8*I*pi/9)/9",
"-sqrt(3)/18 - I/6"
]
}
}