χR(∣CS⟩⊗2) ≥ 3

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^2) >= 3, 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) = 3 with the verified three-term witness, against the product bound 4. Exact throughout: the 36,720 four-qubit stabilizer states are enumerated from the verifier's parametrization as vectors over Z[i] (reduced-row-echelon flats, strictly upper-triangular Q, l in Z4^k; count checked against 2^4 (2+1)(4+1)(8+1)(16+1), pairwise distinct up to phase, and matched one to one against the numerical dictionary as a consistency check nothing rests on). With P = 4|CS⟩⊗2 the component q(s) = <P,P> s - <P,s> P of each state off the target stays in Z[i], and P lies in span(s, t) exactly when q(s) and q(t) are parallel, which is decided by dividing each q(s) by its first nonzero entry in Q(i) and comparing: no two coincide and none vanishes. The design note's numerical search (2026-09-20) found the same with closest approach to parallel 0.763 against the threshold 1 - 1e-6; this certificate removes the margin. Controls: |++++⟩ is reported as rank 1, |CS⟩ ⊗ |++⟩ is spanned by pairs, and on two qubits the pair {|++⟩, CZ|++⟩} is found for |CS⟩. The script also prints χ(CS^3) ≥ 3 by projection monotonicity, (I ⊗ ⟨00|)|CS⟩⊗3 = |CS⟩⊗2 / 2.

Attribution

Vincent Russo · stabrank verify_challenge · exact: no pair of the 36,720 four-qubit stabilizer states spans |CS⟩⊗2, decided in Q(i) by canonicalizing each state's component off the target

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "CS",
  "m": 2,
  "direction": "lower",
  "rank": 3,
  "certificate": {
    "script": "verify_challenge/cert_cs_m2_rank2.py",
    "expect": "CERTIFIED chi(CS^2) >= 3",
    "exact": true
  },
  "provenance": {
    "author": "Vincent Russo",
    "reference": "stabrank verify_challenge",
    "method": "exact: no pair of the 36,720 four-qubit stabilizer states spans |CS>^2, decided in Q(i) by canonicalizing each state's component off the target",
    "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) = 3 with the verified three-term witness, against the product bound 4. Exact throughout: the 36,720 four-qubit stabilizer states are enumerated from the verifier's parametrization as vectors over Z[i] (reduced-row-echelon flats, strictly upper-triangular Q, l in Z_4^k; count checked against 2^4 (2+1)(4+1)(8+1)(16+1), pairwise distinct up to phase, and matched one to one against the numerical dictionary as a consistency check nothing rests on). With P = 4|CS>^2 the component q(s) = <P,P> s - <P,s> P of each state off the target stays in Z[i], and P lies in span(s, t) exactly when q(s) and q(t) are parallel, which is decided by dividing each q(s) by its first nonzero entry in Q(i) and comparing: no two coincide and none vanishes. The design note's numerical search (2026-09-20) found the same with closest approach to parallel 0.763 against the threshold 1 - 1e-6; this certificate removes the margin. Controls: |++++> is reported as rank 1, |CS> (x) |++> is spanned by pairs, and on two qubits the pair {|++>, CZ|++>} is found for |CS>. The script also prints chi(CS^3) >= 3 by projection monotonicity, (I (x) <00|)|CS>^3 = |CS>^2 / 2."
}