χR(∣qubit_sqrtT⟩⊗2) ≤ 3

√T 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

verified   3 stabilizer terms reproduce |qubit_sqrtT>^{ot 2}, checked symbolically

Implied per-copy exponent γ ≤ 0.7925 against a published 0.5000, no improvement.

Compute

1 run; 0.001 CPU-hours; 0.001 h wall-clock; on Apple silicon laptop, one core; direct construction, no search.

Notes

|sqrtT⟩⊗2 lies in the symmetric subspace Sym^2(C^2), spanned by the stabilizer states |00⟩, |11⟩ and (|01⟩ + |10⟩)/√ 2, so χ ≤ 3 at two copies for every single-qubit state. With the exact Galois lower bound 3 the cell is settled, χ(|sqrtT⟩⊗2) = 3, the generic two-copy value (Lovitz and Steffan, arXiv:2110.07781), one more than the T state's 2. The numerical rank-2 exclusion over the 60 two-qubit states (research survey, 2026-10-17) has closest approach 0.986 to the parallel threshold; the lower bound filed is exact instead. Coefficients in Q(zeta_16, √ 2), solved by fit_coeffs in the cyclotomic field; the identity is checked symbolically. Implied γ log_2(3)/2 = 0.7925 against the published 1/2.

Attribution

Vincent Russo · this repository (stabrank-research, round 31 new-orbit survey) · by hand: the symmetric-subspace basis |00⟩, |11⟩, (|01⟩ + |10⟩)/√ 2, coefficients fitted exactly

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "qubit_sqrtT",
  "m": 2,
  "direction": "upper",
  "rank": 3,
  "witness": {
    "terms": [
      {
        "k": 0,
        "x0": [
          0,
          0
        ],
        "W": [],
        "Q": [],
        "l": []
      },
      {
        "k": 0,
        "x0": [
          1,
          1
        ],
        "W": [],
        "Q": [],
        "l": []
      },
      {
        "k": 1,
        "x0": [
          0,
          1
        ],
        "W": [
          [
            1,
            1
          ]
        ],
        "Q": [
          [
            0
          ]
        ],
        "l": [
          0
        ]
      }
    ],
    "coeffs": [
      "1/2",
      "exp(I*pi/4)/2",
      "sqrt(2)*exp(I*pi/8)/2"
    ]
  },
  "provenance": {
    "author": "Vincent Russo",
    "reference": "this repository (stabrank-research, round 31 new-orbit survey)",
    "method": "by hand: the symmetric-subspace basis |00>, |11>, (|01> + |10>)/sqrt 2, coefficients fitted exactly",
    "date": "2026-10-17",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 0.001,
      "wall_clock_hours": 0.001,
      "runs": 1,
      "hardware": "Apple silicon laptop, one core; direct construction, no search"
    }
  },
  "notes": "|sqrtT>^2 lies in the symmetric subspace Sym^2(C^2), spanned by the stabilizer states |00>, |11> and (|01> + |10>)/sqrt 2, so chi <= 3 at two copies for every single-qubit state. With the exact Galois lower bound 3 the cell is settled, chi(|sqrtT>^2) = 3, the generic two-copy value (Lovitz and Steffan, arXiv:2110.07781), one more than the T state's 2. The numerical rank-2 exclusion over the 60 two-qubit states (research survey, 2026-10-17) has closest approach 0.986 to the parallel threshold; the lower bound filed is exact instead. Coefficients in Q(zeta_16, sqrt 2), solved by fit_coeffs in the cyclotomic field; the identity is checked symbolically. Implied gamma log_2(3)/2 = 0.7925 against the published 1/2."
}