verified 18 stabilizer terms reproduce |qubit_H>^{ot 9}, checked symbolically
Implied per-copy exponent γ ≤ 0.4633 against a published 0.3962, no improvement.
1 run; 0.01 CPU-hours; 0.01 h wall-clock; on Apple silicon laptop, one core; direct construction, no search.
χ(|H⟩⊗9) ≤ 18, the first entry for this cell; exponent log_2(18)/9 = 0.4633, above the published log_2(3)/4. Construction: the nine-term glued |cat10⟩ of Qassim, Pashayan, and Gosset (as restated by Kissinger, van de Wetering, and Vilmart) projected onto |0⟩ and onto |1⟩ on the tenth qubit gives |cat9⟩ and |cat_9^→ = (|T⟩⊗9 - |Tperp⟩⊗9)/√ 2 in nine terms each, and |T⟩⊗9 = (|cat9⟩ + |cat_9^→)/√ 2. Every other route through the same machinery (gluing cat_6 with cat_5, the 4-to-3 partial decomposition times the board's m=5 witness, the products T^6 x T^3 and T^7 x T^2) also gives 18 distinct terms; the five routes use 81 distinct stabilizer states spanning a 66-dimensional space, and greedy and randomised pruning inside that pool found nothing shorter than 18 (research/constructions/kvv_cat_m9.py). Built here and verified in exact arithmetic.
this repository · arXiv:2106.07740, arXiv:2202.09202 · structured construction
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"reference": "arXiv:2106.07740, arXiv:2202.09202",
"method": "structured construction",
"date": "2026-09-22",
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"notes": "chi(|H>^9) <= 18, the first entry for this cell; exponent log_2(18)/9 = 0.4633, above the published log_2(3)/4. Construction: the nine-term glued |cat_10> of Qassim, Pashayan, and Gosset (as restated by Kissinger, van de Wetering, and Vilmart) projected onto |0> and onto |1> on the tenth qubit gives |cat_9> and |cat_9^-> = (|T>^9 - |T_perp>^9)/sqrt 2 in nine terms each, and |T>^9 = (|cat_9> + |cat_9^->)/sqrt 2. Every other route through the same machinery (gluing cat_6 with cat_5, the 4-to-3 partial decomposition times the board's m=5 witness, the products T^6 x T^3 and T^7 x T^2) also gives 18 distinct terms; the five routes use 81 distinct stabilizer states spanning a 66-dimensional space, and greedy and randomised pruning inside that pool found nothing shorter than 18 (research/constructions/kvv_cat_m9.py). Built here and verified in exact arithmetic.",
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"coeffs": [
"(-32*sqrt(2 - sqrt(2)) - sqrt(4 - 2*sqrt(2))*(16 + 125*I) - 32*I*sqrt(sqrt(2) + 2) - (125 - 16*I)*sqrt(2*sqrt(2) + 4))/2256",
"((-14241 - 4768*I)*sqrt(sqrt(2) + 2) + (-5768 - 1000*I)*sqrt(4 - 2*sqrt(2)) + (-4768 + 14241*I)*sqrt(2 - sqrt(2)) + (-1000 + 5768*I)*sqrt(2*sqrt(2) + 4))/168072",
"((15241 + 1000*I)*sqrt(2*sqrt(2) + 4) + (-14241 + 4768*I)*sqrt(sqrt(2) + 2) + sqrt(2 - sqrt(2))*(4768 + 14241*I) + (-1000 + 15241*I)*sqrt(4 - 2*sqrt(2)))/168072",
"((-95 - 62*I)*sqrt(sqrt(2) + 2) + (-54 + 8*I)*sqrt(4 - 2*sqrt(2)) + (-62 + 95*I)*sqrt(2 - sqrt(2)) + (8 + 54*I)*sqrt(2*sqrt(2) + 4))/1128",
"-sqrt(1/16 - sqrt(2)/32) + I*sqrt(sqrt(2)/32 + 1/16)",
"sqrt(sqrt(2)/32 + 1/16) - I*sqrt(1/16 - sqrt(2)/32)",
"((87 - 8*I)*sqrt(2*sqrt(2) + 4) + sqrt(2 - sqrt(2))*(62 + 95*I) + sqrt(4 - 2*sqrt(2))*(8 + 87*I) + (-95 + 62*I)*sqrt(sqrt(2) + 2))/1128",
"sqrt(sqrt(2)/32 + 1/16) + I*sqrt(1/16 - sqrt(2)/32)",
"sqrt(sqrt(2)/32 + 1/16) - I*sqrt(1/16 - sqrt(2)/32)",
"(-32*sqrt(2 - sqrt(2)) + (-125 - 16*I)*sqrt(2*sqrt(2) + 4) + 32*sqrt(-2 - sqrt(2)) + (-16 + 125*I)*sqrt(4 - 2*sqrt(2)))/2256",
"((-1000 - 5768*I)*sqrt(2*sqrt(2) + 4) + (-4768 - 14241*I)*sqrt(2 - sqrt(2)) + (-5768 + 1000*I)*sqrt(4 - 2*sqrt(2)) + (-14241 + 4768*I)*sqrt(sqrt(2) + 2))/168072",
"(-sqrt(2)*sqrt(2 - sqrt(2))*(1000 + 15241*I) + sqrt(2 - sqrt(2))*(4768 - 14241*I) - (14241 + 4768*I)*sqrt(sqrt(2) + 2) + sqrt(2)*(15241 - 1000*I)*sqrt(sqrt(2) + 2))/168072",
"((8 - 54*I)*sqrt(2*sqrt(2) + 4) + (-62 - 95*I)*sqrt(2 - sqrt(2)) + (-54 - 8*I)*sqrt(4 - 2*sqrt(2)) + (-95 + 62*I)*sqrt(sqrt(2) + 2))/1128",
"-sqrt(1/16 - sqrt(2)/32) - I*sqrt(sqrt(2)/32 + 1/16)",
"sqrt(sqrt(2)/32 + 1/16) + I*sqrt(1/16 - sqrt(2)/32)",
"((-95 - 62*I)*sqrt(sqrt(2) + 2) + sqrt(4 - 2*sqrt(2))*(8 - 87*I) + sqrt(2 - sqrt(2))*(62 - 95*I) + (87 + 8*I)*sqrt(2*sqrt(2) + 4))/1128",
"sqrt(sqrt(2)/32 + 1/16) - I*sqrt(1/16 - sqrt(2)/32)",
"sqrt(sqrt(2)/32 + 1/16) + I*sqrt(1/16 - sqrt(2)/32)"
]
}
}