verified 3 stabilizer terms reproduce |CS>^{ot 2}, checked symbolically
Implied per-copy exponent γ ≤ 0.7925 against a baseline 1.0000 (single-copy product bound; no published exponent), an improvement.
1 run; 0.01 CPU-hours; 0.01 h wall-clock; on Apple silicon laptop, one core.
|CS⟩⊗2 = (1/2)|0000⟩ - ((1+i)/2)|+i⟩⊗4 - ((1-i)/2)(CZ ⊗ CZ)|-i⟩⊗4, with |+i⟩ = (|0⟩ + i|1⟩)/√ 2 and |-i⟩ its conjugate; the coefficients are in Q(i) and the identity is checked symbolically by the verifier. Found by the design note's pivot search over the 36,720 four-qubit states. There are 16 rank-3 decompositions of |CS⟩⊗2 in all, each one computational basis state plus two full-support states, and they form one orbit under the unitary symmetry group of |CS⟩⊗2 (the 24 two-qubit Cliffords fixing |CS⟩ up to phase on each copy and the swap of the copies, order 1152), reached by the pivot search from three pivot orbits; this is the member with the basis state |0000⟩. With the exact rank-2 exclusion of cert_cs_m2_rank2.py this settles χ(|CS⟩⊗2) = 3, strictly below the product bound 4, so the transversal-number bound 2^tau for hypergraph states is not tight even for two disjoint edges of size two. Implied γ log_2(3)/2 = 0.7925 against the single-copy product baseline 1; no exponent has been published for the orbit, so the site does not count the baseline as beaten. The extent exponent 0.678 is the comparison line for the approximate rank.
Vincent Russo · stabrank docs/notes/new_orbits_design.md · pivot search of rank_exclusion.py over the 36,720 four-qubit stabilizer states (design note, 2026-09-20), then exact refit by fit_coeffs.fit
{
"schema_version": "0.1",
"orbit": "CS",
"m": 2,
"direction": "upper",
"rank": 3,
"witness": {
"terms": [
{
"k": 0,
"x0": [
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"W": [],
"Q": [],
"l": []
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{
"k": 4,
"x0": [
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"W": [
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{
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"Q": [
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"l": [
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}
],
"coeffs": [
"1/2",
"-1/2 - I/2",
"-1/2 + I/2"
]
},
"provenance": {
"author": "Vincent Russo",
"reference": "stabrank docs/notes/new_orbits_design.md",
"method": "pivot search of rank_exclusion.py over the 36,720 four-qubit stabilizer states (design note, 2026-09-20), then exact refit by fit_coeffs.fit",
"date": "2026-09-20",
"github": [
"vprusso"
],
"compute": {
"cpu_hours": 0.01,
"wall_clock_hours": 0.01,
"runs": 1,
"hardware": "Apple silicon laptop, one core"
}
},
"notes": "|CS>^2 = (1/2)|0000> - ((1+i)/2)|+i>^4 - ((1-i)/2)(CZ (x) CZ)|-i>^4, with |+i> = (|0> + i|1>)/sqrt 2 and |-i> its conjugate; the coefficients are in Q(i) and the identity is checked symbolically by the verifier. Found by the design note's pivot search over the 36,720 four-qubit states. There are 16 rank-3 decompositions of |CS>^2 in all, each one computational basis state plus two full-support states, and they form one orbit under the unitary symmetry group of |CS>^2 (the 24 two-qubit Cliffords fixing |CS> up to phase on each copy and the swap of the copies, order 1152), reached by the pivot search from three pivot orbits; this is the member with the basis state |0000>. With the exact rank-2 exclusion of cert_cs_m2_rank2.py this settles chi(|CS>^2) = 3, strictly below the product bound 4, so the transversal-number bound 2^tau for hypergraph states is not tight even for two disjoint edges of size two. Implied gamma log_2(3)/2 = 0.7925 against the single-copy product baseline 1; no exponent has been published for the orbit, so the site does not count the baseline as beaten. The extent exponent 0.678 is the comparison line for the approximate rank."
}