χR(∣qubit_H⟩⊗4) ≥ 3

H-type orbit, 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

reproduced   certificate script asserted: CERTIFIED chi(qubit_H^4) >= 3

Notes

Leaves this cell at 3 ≤ χ ≤ 4. Closes it as a route to the published exponent log_2(3)/4, which would need rank 2 here. The stabilizer states are enumerated by closing the Clifford orbit of |0...0> rather than from a phase parametrisation, since the verifier's witness format uses w2 = -1 and so reaches only the real stabilizer states, missing the Y eigenstates; the count is checked against 2^n prod (2^k + 1). The exclusion rests on a margin rather than exact arithmetic: the closest any two candidate directions come to parallel is 0.76 to 0.78 against a threshold of 1 - 1e-6, so no rounding at the 1e-15 level of these inputs can close the gap. The H-type state at m=2 has a known rank-2 decomposition, which the same code finds as a positive control.

Attribution

this repository · stabrank PR #18 · exhaustion over all pairs of stabilizer states, via the quotient by the target

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "qubit_H",
  "m": 4,
  "direction": "lower",
  "rank": 3,
  "certificate": {
    "script": "verify_challenge/cert_qubit_rank2.py",
    "expect": "CERTIFIED chi(qubit_H^4) >= 3"
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank PR #18",
    "method": "exhaustion over all pairs of stabilizer states, via the quotient by the target",
    "date": "2026-09-12",
    "github": [
      "vprusso"
    ]
  },
  "notes": "Leaves this cell at 3 <= chi <= 4. Closes it as a route to the published exponent log_2(3)/4, which would need rank 2 here. The stabilizer states are enumerated by closing the Clifford orbit of |0...0> rather than from a phase parametrisation, since the verifier's witness format uses w_2 = -1 and so reaches only the real stabilizer states, missing the Y eigenstates; the count is checked against 2^n prod (2^k + 1). The exclusion rests on a margin rather than exact arithmetic: the closest any two candidate directions come to parallel is 0.76 to 0.78 against a threshold of 1 - 1e-6, so no rounding at the 1e-15 level of these inputs can close the gap. The H-type state at m=2 has a known rank-2 decomposition, which the same code finds as a positive control."
}