χR(∣CS⟩⊗1) ≥ 2

CS orbit, two-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

verified   certificate script asserted: CERTIFIED chi(CS^1) >= 2, by an argument declared exact throughout (no floating-point margin)

Compute

1 run; 0.005 CPU-hours; 0.005 h wall-clock; on Apple silicon laptop, one core; about 15 s for the certificate; claude-fable-5-1, wrote the exact certificate.

Notes

Settles χ(|CS⟩) = 2 with the two-term witness. |CS⟩ is not a stabilizer state: its support is all of F2^2 but the phase i^(x1 x2) is not a Z4-valued quadratic form with even cross terms (the cross term is odd), and the certificate decides this as the rank-1 case of its exact pair search: every two-qubit stabilizer state is written down from the verifier's parametrization as a vector over Z[i] (60 states, checked against 2^2 (2+1)(4+1) and pairwise distinct up to phase), and none has a vanishing component off |CS⟩. Exact throughout; no floating-point margin stands under the bound. Controls: |++⟩ is reported as rank 1, and the pair {|++⟩, CZ|++⟩} is among the 15 pairs of two-qubit stabilizer states that span |CS⟩.

Attribution

Vincent Russo · stabrank verify_challenge · exact: no two-qubit stabilizer state is parallel to |CS⟩, decided in Q(i) over the 60 states enumerated from the parametrization

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "CS",
  "m": 1,
  "direction": "lower",
  "rank": 2,
  "certificate": {
    "script": "verify_challenge/cert_cs_m2_rank2.py",
    "expect": "CERTIFIED chi(CS^1) >= 2",
    "exact": true
  },
  "provenance": {
    "author": "Vincent Russo",
    "reference": "stabrank verify_challenge",
    "method": "exact: no two-qubit stabilizer state is parallel to |CS>, decided in Q(i) over the 60 states enumerated from the parametrization",
    "date": "2026-10-21",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 0.005,
      "wall_clock_hours": 0.005,
      "runs": 1,
      "hardware": "Apple silicon laptop, one core; about 15 s for the certificate",
      "llm": [
        {
          "model": "claude-fable-5-1",
          "role": "wrote the exact certificate"
        }
      ]
    }
  },
  "notes": "Settles chi(|CS>) = 2 with the two-term witness. |CS> is not a stabilizer state: its support is all of F_2^2 but the phase i^(x1 x2) is not a Z_4-valued quadratic form with even cross terms (the cross term is odd), and the certificate decides this as the rank-1 case of its exact pair search: every two-qubit stabilizer state is written down from the verifier's parametrization as a vector over Z[i] (60 states, checked against 2^2 (2+1)(4+1) and pairwise distinct up to phase), and none has a vanishing component off |CS>. Exact throughout; no floating-point margin stands under the bound. Controls: |++> is reported as rank 1, and the pair {|++>, CZ|++>} is among the 15 pairs of two-qubit stabilizer states that span |CS>."
}