χ_R(∣cat⟩^⊗3) ≥ 2

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.

← back to the board

Verification

verified certificate script asserted: CERTIFIED chi(cat_3) >= 2, by an argument declared exact throughout (no floating-point margin)

Notes

χ(cat_3) ≥ 2: |cat₃⟩ = (|000⟩ + i(|011⟩ + |101⟩ + |110⟩))/2 is not a stabilizer state. Qassim, Pashayan, and Gosset (arXiv:2106.07740, appendix) show it through the Pauli spectrum, ⟨cat_3|XXI|cat₃⟩ = 1/2; the certificate decides the same fact in integer arithmetic on the exact amplitudes: the support is the affine subspace of even-weight strings, but the phase i^{|x|/2} on it is not a Z₄ quadratic form with even cross terms (at x = 011, x' = 101 the ratio f(x + x') f(0)/(f(x) f(x')) is -i). With bounds/cat-m3-upper-2.json the cell is exact, χ(cat_3) = 2, as in their Table 1.

Attribution

this repository · stabrank verify_challenge · exact: the state is not a stabilizer state (its phase on the even-weight strings is not a Z₄ quadratic form)

Submission

JSON

{
  "schema_version": "0.1",
  "orbit": "cat",
  "m": 3,
  "direction": "lower",
  "rank": 2,
  "certificate": {
    "script": "verify_challenge/cert_family_rank1.py",
    "exact": true,
    "expect": "CERTIFIED chi(cat_3) >= 2"
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank verify_challenge",
    "date": "2026-09-24",
    "github": [
      "vprusso"
    ],
    "method": "exact: the state is not a stabilizer state (its phase on the even-weight strings is not a Z_4 quadratic form)"
  },
  "notes": "chi(cat_3) >= 2: |cat_3> = (|000> + i(|011> + |101> + |110>))/2 is not a stabilizer state. Qassim, Pashayan, and Gosset (arXiv:2106.07740, appendix) show it through the Pauli spectrum, <cat_3|XXI|cat_3> = 1/2; the certificate decides the same fact in integer arithmetic on the exact amplitudes: the support is the affine subspace of even-weight strings, but the phase i^{|x|/2} on it is not a Z_4 quadratic form with even cross terms (at x = 011, x' = 101 the ratio f(x + x') f(0)/(f(x) f(x')) is -i). With bounds/cat-m3-upper-2.json the cell is exact, chi(cat_3) = 2, as in their Table 1."
}