χR(∣qubit_H⟩⊗4) ≥ 4

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.

← back to the board

Verification

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

Compute

1 run; 0.3 CPU-hours; 0.08 h wall-clock; on Apple silicon laptop, 8 worker processes.

Notes

Settles χ(|H⟩⊗4) = 4 with the verified rank-4 decomposition, both directions machine-checked, and reproduces the value of Bravyi, Smith, and Smolin from scratch. Every rank-3 triple over the 36720 four-qubit stabilizer states is excluded: 8.69 million collinear-in-quotient candidates arise and every one is three states of a single two-dimensional stabilizer subspace, whose span cannot contain the target once rank 2 is excluded in the same pass; no independent collinear triple exists at all. The exclusion is numerical and rests on a margin: the closest any two key-adjacent non-parallel quotient directions come to parallel is 0.9877 against a threshold of 1 - 1e-6, and the certificate refuses to print its claim if that clearance falls under 0.01. The H-type state at m=3, whose rank-3 decomposition is verified on the board, is the positive control.

Attribution

this repository · stabrank rank_exclusion.py · exhaustion over all triples of 4-qubit stabilizer states, via a pivot and two quotients

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "qubit_H",
  "m": 4,
  "direction": "lower",
  "rank": 4,
  "certificate": {
    "script": "verify_challenge/cert_qubit_h_m4_rank3.py",
    "expect": "CERTIFIED chi(qubit_H^4) >= 4"
  },
  "provenance": {
    "author": "this repository",
    "reference": "stabrank rank_exclusion.py",
    "method": "exhaustion over all triples of 4-qubit stabilizer states, via a pivot and two quotients",
    "date": "2026-09-16",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 0.3,
      "wall_clock_hours": 0.08,
      "runs": 1,
      "hardware": "Apple silicon laptop, 8 worker processes"
    }
  },
  "notes": "Settles chi(|H>^4) = 4 with the verified rank-4 decomposition, both directions machine-checked, and reproduces the value of Bravyi, Smith, and Smolin from scratch. Every rank-3 triple over the 36720 four-qubit stabilizer states is excluded: 8.69 million collinear-in-quotient candidates arise and every one is three states of a single two-dimensional stabilizer subspace, whose span cannot contain the target once rank 2 is excluded in the same pass; no independent collinear triple exists at all. The exclusion is numerical and rests on a margin: the closest any two key-adjacent non-parallel quotient directions come to parallel is 0.9877 against a threshold of 1 - 1e-6, and the certificate refuses to print its claim if that clearance falls under 0.01. The H-type state at m=3, whose rank-3 decomposition is verified on the board, is the positive control."
}