χR(∣cat⟩⊗5) ≥ 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.

← back to the board

Verification

verified   certificate script asserted: CERTIFIED chi(cat_5) >= 3, by an argument declared exact throughout (no floating-point margin)

Compute

3 runs; 0.01 CPU-hours; 0.01 h wall-clock; on Apple silicon laptop, one core; claude-opus-5-5, designed, wrote and ran the certificate and its cross-checks.

Notes

Machine check of Lemma 3 in the appendix of Qassim, Pashayan, and Gosset (arXiv:2106.07740), χ(cat_5) > 2, which they prove by a computer comparison of Pauli spectra; the rank is theirs, the check is new. With bounds/cat-m5-upper-3.json the cell is exact, χ(cat_5) = 3, as in their Table 1. Method: |cat5⟩ lies in the stabilizer code Z^{(x)5} = +1, and projecting a decomposition onto that code keeps each term a stabilizer state or zero; the CNOT chain onto the last qubit identifies the code with four qubits, so χ(cat_5) is the rank of the four-qubit state c5(y) = cat_5(y, parity(y)), whose entries are fourth roots of unity. All 36,720 four-qubit stabilizer states are generated exactly as Gaussian-integer vectors (count asserted). With q(s) = s - s0 t, exact over Z[i], the target lies in the span of two states iff their quotients are parallel, decided by exact reduced Gaussian-rational keys; no floating point enters the decision. Controls in the script: compressed cat_4 (χ = 2) shows its 15 rank-2 classes, a stabilizer target is flagged as rank 1, and the closed form of |catm⟩ matches stabrank_verify.cat_vector exactly for m = 4, 5, 6. About 8 s on one core.

Attribution

Michel de Messieres · stabrank verify_challenge · exact rank-2 exclusion over the 36,720 four-qubit stabilizer states after compressing |cat5⟩ into the even-weight code Z^5 = +1

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "cat",
  "m": 5,
  "direction": "lower",
  "rank": 3,
  "certificate": {
    "script": "verify_challenge/cert_cat_m5_rank2.py",
    "exact": true,
    "expect": "CERTIFIED chi(cat_5) >= 3"
  },
  "provenance": {
    "author": "Michel de Messieres",
    "reference": "stabrank verify_challenge",
    "date": "2026-10-16",
    "github": [
      "MicheldeMessieres"
    ],
    "method": "exact rank-2 exclusion over the 36,720 four-qubit stabilizer states after compressing |cat_5> into the even-weight code Z^5 = +1",
    "compute": {
      "cpu_hours": 0.01,
      "wall_clock_hours": 0.01,
      "runs": 3,
      "hardware": "Apple silicon laptop, one core",
      "llm": [
        {
          "model": "claude-opus-5-5",
          "role": "designed, wrote and ran the certificate and its cross-checks"
        }
      ]
    }
  },
  "notes": "Machine check of Lemma 3 in the appendix of Qassim, Pashayan, and Gosset (arXiv:2106.07740), chi(cat_5) > 2, which they prove by a computer comparison of Pauli spectra; the rank is theirs, the check is new. With bounds/cat-m5-upper-3.json the cell is exact, chi(cat_5) = 3, as in their Table 1. Method: |cat_5> lies in the stabilizer code Z^{(x)5} = +1, and projecting a decomposition onto that code keeps each term a stabilizer state or zero; the CNOT chain onto the last qubit identifies the code with four qubits, so chi(cat_5) is the rank of the four-qubit state c_5(y) = cat_5(y, parity(y)), whose entries are fourth roots of unity. All 36,720 four-qubit stabilizer states are generated exactly as Gaussian-integer vectors (count asserted). With q(s) = s - s_0 t, exact over Z[i], the target lies in the span of two states iff their quotients are parallel, decided by exact reduced Gaussian-rational keys; no floating point enters the decision. Controls in the script: compressed cat_4 (chi = 2) shows its 15 rank-2 classes, a stabilizer target is flagged as rank 1, and the closed form of |cat_m> matches stabrank_verify.cat_vector exactly for m = 4, 5, 6. About 8 s on one core."
}