verified 2 stabilizer terms reproduce |CS>^{ot 1}, checked symbolically
Implied per-copy exponent γ ≤ 1.0000 against a baseline 1.0000 (single-copy product bound; no published exponent), no improvement.
1 run; 0.001 CPU-hours; 0.001 h wall-clock; on Apple silicon laptop, one core; direct construction, no search.
|CS⟩ = CS|++⟩ = (|00⟩ + |01⟩ + |10⟩ + i|11⟩)/2 is the magic state of the two-qubit controlled-S gate, the first width-2 orbit on the board: m copies live on 2m qubits and γ = log_2(rank)/m is still per copy. CS = ((1+i)/2) I + ((1-i)/2) CZ, so |CS⟩ = ((1+i)/2)|++⟩ + ((1-i)/2) CZ|++⟩, two stabilizer terms with coefficients in Q(i); the identity is checked symbolically by the verifier. With the exact rank-1 exclusion of cert_cs_m2_rank2.py this settles χ(|CS⟩) = 2. No exponent has been published for the CS orbit, so under the board's convention its baseline is the single-copy product bound log_2 χ(|CS⟩) = 1, labeled as such on every page. Comparison lines from the literature: the extent exponent log_2(8/5) = 0.678 (F(CS) = 5/8 over the 60 two-qubit stabilizer states and xi(CS) = 8/5 by the Clifford-magic-state theorem of Bravyi, Browne, Calpin, Campbell, Gosset, and Howard, arXiv:1808.00128; an approximate-rank exponent, not a bound on χ) and the T-synthesis exponent 3 log_2(3)/4 = 1.19 from three T gates per CS.
Vincent Russo · stabrank docs/notes/new_orbits_design.md · by hand: CS = ((1+i)/2) I + ((1-i)/2) CZ applied to |++⟩, coefficients fitted exactly by fit_coeffs.fit
{
"schema_version": "0.1",
"orbit": "CS",
"m": 1,
"direction": "upper",
"rank": 2,
"witness": {
"terms": [
{
"k": 2,
"x0": [
0,
0
],
"W": [
[
1,
0
],
[
0,
1
]
],
"Q": [
[
0,
0
],
[
0,
0
]
],
"l": [
0,
0
]
},
{
"k": 2,
"x0": [
0,
0
],
"W": [
[
1,
0
],
[
0,
1
]
],
"Q": [
[
0,
1
],
[
0,
0
]
],
"l": [
0,
0
]
}
],
"coeffs": [
"1/2 + I/2",
"1/2 - I/2"
]
},
"provenance": {
"author": "Vincent Russo",
"reference": "stabrank docs/notes/new_orbits_design.md",
"method": "by hand: CS = ((1+i)/2) I + ((1-i)/2) CZ applied to |++>, coefficients fitted exactly by fit_coeffs.fit",
"date": "2026-09-20",
"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": "|CS> = CS|++> = (|00> + |01> + |10> + i|11>)/2 is the magic state of the two-qubit controlled-S gate, the first width-2 orbit on the board: m copies live on 2m qubits and gamma = log_2(rank)/m is still per copy. CS = ((1+i)/2) I + ((1-i)/2) CZ, so |CS> = ((1+i)/2)|++> + ((1-i)/2) CZ|++>, two stabilizer terms with coefficients in Q(i); the identity is checked symbolically by the verifier. With the exact rank-1 exclusion of cert_cs_m2_rank2.py this settles chi(|CS>) = 2. No exponent has been published for the CS orbit, so under the board's convention its baseline is the single-copy product bound log_2 chi(|CS>) = 1, labeled as such on every page. Comparison lines from the literature: the extent exponent log_2(8/5) = 0.678 (F(CS) = 5/8 over the 60 two-qubit stabilizer states and xi(CS) = 8/5 by the Clifford-magic-state theorem of Bravyi, Browne, Calpin, Campbell, Gosset, and Howard, arXiv:1808.00128; an approximate-rank exponent, not a bound on chi) and the T-synthesis exponent 3 log_2(3)/4 = 1.19 from three T gates per CS."
}