verified 12 stabilizer terms reproduce |qubit_sqrtT>^{ot 6}, checked symbolically
Implied per-copy exponent γ ≤ 0.5975 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 qubit_sqrtT-m5-upper-6.json, qubit_sqrtT-m1-upper-2.json: χ(|qubitsqrtT⟩⊗5) ≤ 6 x χ(|qubitsqrtT⟩⊗1) ≤ 2 gives χ(|qubitsqrtT⟩⊗6) ≤ 12, exponent 0.5975. 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": "qubit_sqrtT",
"m": 6,
"direction": "upper",
"rank": 12,
"provenance": {
"author": "this repository",
"reference": "stabrank research/constructions/product_witness.py",
"method": "structured construction",
"date": "2026-10-23",
"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 qubit_sqrtT-m5-upper-6.json, qubit_sqrtT-m1-upper-2.json: chi(|qubit_sqrtT>^5) <= 6 x chi(|qubit_sqrtT>^1) <= 2 gives chi(|qubit_sqrtT>^6) <= 12, exponent 0.5975. 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": 0,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [],
"Q": [],
"l": []
},
{
"k": 0,
"x0": [
0,
0,
0,
0,
0,
1
],
"W": [],
"Q": [],
"l": []
},
{
"k": 0,
"x0": [
1,
1,
1,
1,
1,
0
],
"W": [],
"Q": [],
"l": []
},
{
"k": 0,
"x0": [
1,
1,
1,
1,
1,
1
],
"W": [],
"Q": [],
"l": []
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
]
],
"Q": [
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
]
],
"l": [
0,
0,
0,
0,
0
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0,
1
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
]
],
"Q": [
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
]
],
"l": [
0,
0,
0,
0,
0
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
]
],
"Q": [
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
]
],
"l": [
2,
2,
2,
2,
2
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0,
1
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
]
],
"Q": [
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
]
],
"l": [
2,
2,
2,
2,
2
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
]
],
"Q": [
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
]
],
"l": [
1,
1,
1,
1,
1
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0,
1
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
]
],
"Q": [
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
]
],
"l": [
1,
1,
1,
1,
1
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0,
0
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
]
],
"Q": [
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
]
],
"l": [
3,
3,
3,
3,
3
]
},
{
"k": 5,
"x0": [
0,
0,
0,
0,
0,
1
],
"W": [
[
1,
0,
0,
0,
0,
0
],
[
0,
1,
0,
0,
0,
0
],
[
0,
0,
1,
0,
0,
0
],
[
0,
0,
0,
1,
0,
0
],
[
0,
0,
0,
0,
1,
0
]
],
"Q": [
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
],
[
0,
0,
0,
0,
0
]
],
"l": [
3,
3,
3,
3,
3
]
}
],
"coeffs": [
"1/8 - I/8",
"(-I*exp(I*pi/8) + exp(I*pi/8))/8",
"(-exp(I*pi/8) + exp(5*I*pi/8))/8",
"(-exp(I*pi/4) + exp(3*I*pi/4))/8",
"(sqrt(2)*exp(I*pi/8) + sqrt(2)*exp(I*pi/4) + sqrt(2)*exp(3*I*pi/8) + sqrt(2)*I)/8",
"(sqrt(2)*exp(I*pi/4) + sqrt(2)*I*exp(I*pi/8) + sqrt(2)*exp(3*I*pi/8) + sqrt(2)*I)/8",
"(-sqrt(2)*exp(3*I*pi/8) - sqrt(2)*exp(I*pi/8) + sqrt(2)*exp(I*pi/4) + sqrt(2)*I)/8",
"(-sqrt(2)*I - sqrt(2)*exp(I*pi/4) + sqrt(2)*I*exp(I*pi/8) + sqrt(2)*exp(3*I*pi/8))/8",
"(-sqrt(2)*exp(5*I*pi/8) - sqrt(2)*exp(I*pi/4) + sqrt(2)*exp(7*I*pi/8) + sqrt(2)*I)/8",
"(-sqrt(2) - sqrt(2)*exp(3*I*pi/8) - sqrt(2)*exp(3*I*pi/4) + sqrt(2)*I*exp(I*pi/8))/8",
"(-sqrt(2)*exp(I*pi/4) - sqrt(2)*exp(7*I*pi/8) + sqrt(2)*exp(5*I*pi/8) + sqrt(2)*I)/8",
"(sqrt(2) - sqrt(2)*exp(3*I*pi/8) + sqrt(2)*exp(3*I*pi/4) + sqrt(2)*I*exp(I*pi/8))/8"
]
}
}