verified 5 stabilizer terms reproduce |qubit_sqrtT>^{ot 4}, checked symbolically
Implied per-copy exponent γ ≤ 0.5805 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.
|0⟩⊗4, |1⟩⊗4, |+⟩⊗4, |→^4 and |+i⟩⊗4 span the five-dimensional symmetric subspace (the six product stabilizer states |0⟩⊗n, |1⟩⊗n, |+→^n, |+-i⟩⊗n span Sym^n for n ≤ 5 and only a six-dimensional subspace beyond), so χ ≤ 5 at four copies for every single-qubit state; this is the generic value chi_4 = 5 of Lovitz and Steffan (arXiv:2110.07781, Section 1.3). Puts the cell at 4 ≤ χ(|sqrtT⟩⊗4) ≤ 5. Whether 4 is attained is open: the research survey's cap-4 MILP over the 36,720 four-qubit words did not finish in 600 s, and the rank-4 pivot-pair search of slice_lift.all_decompositions is the tool. Implied γ log_2(5)/4 = 0.5805 against the published 1/2. For comparison χ(|T⟩⊗4) = 3 (edge orbit, lean tier) and χ(|H⟩⊗4) = 4.
Vincent Russo · this repository (stabrank-research, round 31 new-orbit survey) · by hand: five product stabilizer states spanning Sym^4(C^2), coefficients fitted exactly in Q(zeta_16)
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"coeffs": [
"1/4 - exp(I*pi/4)/4 - exp(7*I*pi/8)/4 + exp(5*I*pi/8)/4",
"-exp(I*pi/4)/4 - exp(7*I*pi/8)/4 + exp(5*I*pi/8)/4 + I/4",
"-exp(5*I*pi/8)/4 + exp(7*I*pi/8)/4 + exp(I*pi/8)/4 + exp(3*I*pi/8)/4 + exp(I*pi/4)/2",
"-exp(3*I*pi/8)/4 - exp(5*I*pi/8)/4 - exp(I*pi/8)/4 + exp(7*I*pi/8)/4 + exp(I*pi/4)/2",
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"provenance": {
"author": "Vincent Russo",
"reference": "this repository (stabrank-research, round 31 new-orbit survey)",
"method": "by hand: five product stabilizer states spanning Sym^4(C^2), coefficients fitted exactly in Q(zeta_16)",
"date": "2026-10-17",
"github": [
"vprusso"
],
"compute": {
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"runs": 1,
"hardware": "Apple silicon laptop, one core; direct construction, no search"
}
},
"notes": "|0>^4, |1>^4, |+>^4, |->^4 and |+i>^4 span the five-dimensional symmetric subspace (the six product stabilizer states |0>^n, |1>^n, |+->^n, |+-i>^n span Sym^n for n <= 5 and only a six-dimensional subspace beyond), so chi <= 5 at four copies for every single-qubit state; this is the generic value chi_4 = 5 of Lovitz and Steffan (arXiv:2110.07781, Section 1.3). Puts the cell at 4 <= chi(|sqrtT>^4) <= 5. Whether 4 is attained is open: the research survey's cap-4 MILP over the 36,720 four-qubit words did not finish in 600 s, and the rank-4 pivot-pair search of slice_lift.all_decompositions is the tool. Implied gamma log_2(5)/4 = 0.5805 against the published 1/2. For comparison chi(|T>^4) = 3 (edge orbit, lean tier) and chi(|H>^4) = 4."
}