χR(∣cat⟩⊗6) ≥ 3

magic cat orbit, m qubits.

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

verified   certificate script asserted: CERTIFIED chi(cat_6) >= 3, by an argument declared exact throughout (no floating-point margin)

Compute

0 runs; 0 CPU-hours; 0 h wall-clock; on none beyond the m=5 certificate, which this cell re-runs.

Notes

χ(cat_6) ≥ χ(cat_5) ≥ 3. (I ⊗ ⟨0|)|cat6⟩ = 2^{-1/2}|cat5⟩, checked by the certificate on the exact amplitudes, and ⟨0| on one qubit carries a stabilizer decomposition to one with no more terms (the projection lemma of cert_s_m4_from_m3.py), so the rank is non-decreasing in m; χ(cat_5) ≥ 3 is the exact exclusion of bounds/cat-m5-lower-3.json, which the same script re-runs (about 8 s). Qassim, Pashayan, and Gosset (arXiv:2106.07740) give the same argument from their Lemma 3. With bounds/cat-m6-upper-3.json the cell is exact, χ(cat_6) = 3, as in their Table 1 and in the text after Theorem 1 ("the stabilizer rank of |cat6⟩ is exactly 3").

Attribution

Michel de Messieres · stabrank verify_challenge · projection monotonicity applied to the exact rank-2 exclusion at m=5

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "cat",
  "m": 6,
  "direction": "lower",
  "rank": 3,
  "certificate": {
    "script": "verify_challenge/cert_cat_m5_rank2.py",
    "exact": true,
    "expect": "CERTIFIED chi(cat_6) >= 3"
  },
  "provenance": {
    "author": "Michel de Messieres",
    "reference": "stabrank verify_challenge",
    "date": "2026-10-16",
    "github": [
      "MicheldeMessieres"
    ],
    "method": "projection monotonicity applied to the exact rank-2 exclusion at m=5",
    "compute": {
      "cpu_hours": 0,
      "wall_clock_hours": 0,
      "runs": 0,
      "hardware": "none beyond the m=5 certificate, which this cell re-runs"
    }
  },
  "notes": "chi(cat_6) >= chi(cat_5) >= 3. (I (x) <0|)|cat_6> = 2^{-1/2}|cat_5>, checked by the certificate on the exact amplitudes, and <0| on one qubit carries a stabilizer decomposition to one with no more terms (the projection lemma of cert_s_m4_from_m3.py), so the rank is non-decreasing in m; chi(cat_5) >= 3 is the exact exclusion of bounds/cat-m5-lower-3.json, which the same script re-runs (about 8 s). Qassim, Pashayan, and Gosset (arXiv:2106.07740) give the same argument from their Lemma 3. With bounds/cat-m6-upper-3.json the cell is exact, chi(cat_6) = 3, as in their Table 1 and in the text after Theorem 1 (\"the stabilizer rank of |cat_6> is exactly 3\")."
}