χ_R(∣qubit_H⟩^⊗2) ≥ 2

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

lean qubit_h_m2_stabRankP_gt_one in LeanProofs.QubitM2Lower, machine-checked by Lean

Lean proof

This bound is backed by a machine-checked theorem: qubit_h_m2_stabRankP_gt_one in LeanProofs.QubitM2Lower. The recorded lake build confirms it compiles.

Notes

Settles χ(H^2) = 2 with the Lean upper bound qubit_h_m2_stabRankP_le_two. A stabilizer state has one modulus on its support; |H⟩^⊗2 has moduli cos^2(π/8) and cos(π/8) sin(π/8), which differ. Also proved in Lean against stabRankP 2: QubitM2Lower.qubit_h_m2_stabRankP_gt_one (every nonzero entry of a stabilizer state has the same fourth power, and equal fourth powers at 00 and 01 would force √2 = -4/3).

Attribution

this repository · stabrank verify_challenge · exact: the state is not a stabilizer state (two distinct amplitude moduli)

Submission

JSON

{
  "schema_version": "0.1",
  "orbit": "qubit_H",
  "m": 2,
  "direction": "lower",
  "rank": 2,
  "certificate": {
    "script": "verify_challenge/cert_rank1_moduli.py",
    "expect": "CERTIFIED chi(qubit_H^2) >= 2",
    "exact": true
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank verify_challenge",
    "method": "exact: the state is not a stabilizer state (two distinct amplitude moduli)",
    "date": "2026-09-20",
    "github": [
      "vprusso"
    ]
  },
  "notes": "Settles chi(H^2) = 2 with the Lean upper bound qubit_h_m2_stabRankP_le_two. A stabilizer state has one modulus on its support; |H>^2 has moduli cos^2(pi/8) and cos(pi/8) sin(pi/8), which differ. Also proved in Lean against stabRankP 2: QubitM2Lower.qubit_h_m2_stabRankP_gt_one (every nonzero entry of a stabilizer state has the same fourth power, and equal fourth powers at 00 and 01 would force sqrt2 = -4/3).",
  "lean": {
    "module": "LeanProofs.QubitM2Lower",
    "theorem": "qubit_h_m2_stabRankP_gt_one"
  }
}