χR(∣H3⟩⊗4) ≥ 5

H₃ orbit, qutrit.

Participate

Run it yourself, or point a coding agent at it.

git clone https://github.com/unitaryfoundation/stabrank
cd stabrank
# describe your decomposition in bounds/mybound.json
make fit BOUND=bounds/mybound.json ARGS=--write
make verify BOUND=bounds/mybound.json

fit solves for the exact coefficients, or tells you no exact combination of your terms reproduces the target. verify rebuilds the identity in exact arithmetic. If it passes, open a pull request adding the file and the leaderboard picks it up on the next build.

A bound the pipeline cannot check is recorded as cited and never takes a record.

← back to the board

Verification

reproduced · 3600 s   certificate script asserted: CERTIFIED chi(H3^4) >= 5 within a declared 3600s budget

Compute

1 run; 1.4 CPU-hours; 0.35 h wall-clock; on Apple silicon laptop, four workers at low priority under a load average near 12: 1183 s for the enumeration and lift.

Notes

χ(H3^3) = 4 exactly (Lean witness and rank-3 exclusion). A rank-4 decomposition of |H3⟩⊗4 sliced along one qutrit gives a rank-4 decomposition of |H3⟩⊗3 whose terms are related slice to slice by a Pauli and cube roots of unity. |H3⟩⊗3 has one rank-4 decomposition up to its unitary symmetries (group of order 384, 170 pivot orbits) and no Pauli assignment satisfies the other two slices. Supersedes the rank-2 exclusion at m=4 (χ ≥ 3). The enumeration runs in the compiled pivot-pair kernel with the stabilizer reductions; it is over the default budget because the symmetry group is small, so the certificate declares 3600 s. Puts the cell at 5 ≤ χ ≤ 8.

Attribution

this repository · stabrank slice_lift.py · slice-and-lift: the single rank-4 decomposition of |H3⟩⊗3 up to symmetry does not extend to |H3⟩⊗4

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "H3",
  "m": 4,
  "direction": "lower",
  "rank": 5,
  "certificate": {
    "script": "verify_challenge/cert_h3_m4_lift.py",
    "expect": "CERTIFIED chi(H3^4) >= 5",
    "budget_s": 3600
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank slice_lift.py",
    "method": "slice-and-lift: the single rank-4 decomposition of |H3>^3 up to symmetry does not extend to |H3>^4",
    "date": "2026-09-18",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 1.4,
      "wall_clock_hours": 0.35,
      "runs": 1,
      "hardware": "Apple silicon laptop, four workers at low priority under a load average near 12: 1183 s for the enumeration and lift"
    }
  },
  "notes": "chi(H3^3) = 4 exactly (Lean witness and rank-3 exclusion). A rank-4 decomposition of |H3>^4 sliced along one qutrit gives a rank-4 decomposition of |H3>^3 whose terms are related slice to slice by a Pauli and cube roots of unity. |H3>^3 has one rank-4 decomposition up to its unitary symmetries (group of order 384, 170 pivot orbits) and no Pauli assignment satisfies the other two slices. Supersedes the rank-2 exclusion at m=4 (chi >= 3). The enumeration runs in the compiled pivot-pair kernel with the stabilizer reductions; it is over the default budget because the symmetry group is small, so the certificate declares 3600 s. Puts the cell at 5 <= chi <= 8."
}