χR(∣qubit_H⟩⊗5) ≥ 5

H-type orbit, qubit.

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.

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Verification

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

Compute

1 run; 1.9 CPU-hours; 0.46 h wall-clock; on Apple silicon laptop, four workers at low priority under a load average near 12: 1649 s for the enumeration and lift.

Notes

χ(H^4) = 4 exactly (Lean tensor-square witness and the rank-3 exclusion over 36720 states). A rank-4 decomposition of |H⟩⊗5 sliced along one qubit has slice 0 a rank-4 decomposition of |H⟩⊗4 and slice 1 a Pauli image of it, term by term, times a fourth root of unity. |H⟩⊗4 has 30 rank-4 decompositions up to its unitary symmetries (group of order 384, 246 pivot orbits) and no Pauli assignment satisfies the slice-1 equation for any of them. Bravyi, Smith, and Smolin's rank-6 decomposition is the upper bound, so the cell is 5 ≤ χ ≤ 6; whether χ(H^5) is 5 or 6 is open, and a rank-5 witness would give exponent log2(5)/5 = 0.464. The certificate declares the 3600 s budget: the qubit H-type symmetry group is small and the enumeration ran 1649 s on four loaded cores.

Attribution

this repository · stabrank slice_lift.py · slice-and-lift: none of the 30 rank-4 decompositions of |H⟩⊗4 up to symmetry extends to |H⟩⊗5

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "qubit_H",
  "m": 5,
  "direction": "lower",
  "rank": 5,
  "certificate": {
    "script": "verify_challenge/cert_qubit_h_m5_lift.py",
    "expect": "CERTIFIED chi(qubit_H^5) >= 5",
    "budget_s": 3600
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank slice_lift.py",
    "method": "slice-and-lift: none of the 30 rank-4 decompositions of |H>^4 up to symmetry extends to |H>^5",
    "date": "2026-09-18",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 1.9,
      "wall_clock_hours": 0.46,
      "runs": 1,
      "hardware": "Apple silicon laptop, four workers at low priority under a load average near 12: 1649 s for the enumeration and lift"
    }
  },
  "notes": "chi(H^4) = 4 exactly (Lean tensor-square witness and the rank-3 exclusion over 36720 states). A rank-4 decomposition of |H>^5 sliced along one qubit has slice 0 a rank-4 decomposition of |H>^4 and slice 1 a Pauli image of it, term by term, times a fourth root of unity. |H>^4 has 30 rank-4 decompositions up to its unitary symmetries (group of order 384, 246 pivot orbits) and no Pauli assignment satisfies the slice-1 equation for any of them. Bravyi, Smith, and Smolin's rank-6 decomposition is the upper bound, so the cell is 5 <= chi <= 6; whether chi(H^5) is 5 or 6 is open, and a rank-5 witness would give exponent log2(5)/5 = 0.464. The certificate declares the 3600 s budget: the qubit H-type symmetry group is small and the enumeration ran 1649 s on four loaded cores."
}