χR(∣qubit_sqrtT⟩⊗1) ≤ 2

√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   2 stabilizer terms reproduce |qubit_sqrtT>^{ot 1}, checked symbolically

Implied per-copy exponent γ ≤ 1.0000 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⟩ = (|0⟩ + e^{i π/8}|1⟩)/√ 2 = Z(π/8)|+⟩, the magic state of the square root of the T gate, at the fourth level of the Clifford hierarchy. It is not Clifford-equivalent to the H-type (edge) or T-type (face) orbits: its Bloch vector (cos(π/8), sin(π/8), 0) is neither an edge nor a face center of the stabilizer octahedron, and the Clifford group acts on Bloch vectors by signed coordinate permutations. Two basis states span it, and with the exact Galois lower bound of cert_sqrtt_galois.py this settles χ(|sqrtT⟩) = 2. Baseline under the board's rule: 1/2, published, from Theorem 2 of Qassim, Pashayan, and Gosset (arXiv:2106.07740), χ(R_theta^m) = O(2^{m/2}) for every angle theta (explicitly χ(R_theta^(24t+6)) ≤ 2^(12t+4)). The question the orbit asks is whether a finer phase is dearer per copy than the T state's log_2(3)/4 = 0.3963; the cells that would beat 1/2 are χ(|sqrtT⟩⊗5) ≤ 5 (0.4644) and χ(|sqrtT⟩⊗6) ≤ 7 (0.4680).

Attribution

Vincent Russo · this repository (stabrank-research, round 31 new-orbit survey) · by hand: the computational basis, coefficients fitted exactly

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "qubit_sqrtT",
  "m": 1,
  "direction": "upper",
  "rank": 2,
  "witness": {
    "terms": [
      {
        "k": 0,
        "x0": [
          0
        ],
        "W": [],
        "Q": [],
        "l": []
      },
      {
        "k": 0,
        "x0": [
          1
        ],
        "W": [],
        "Q": [],
        "l": []
      }
    ],
    "coeffs": [
      "sqrt(2)/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 computational basis, 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> = (|0> + e^{i pi/8}|1>)/sqrt 2 = Z(pi/8)|+>, the magic state of the square root of the T gate, at the fourth level of the Clifford hierarchy. It is not Clifford-equivalent to the H-type (edge) or T-type (face) orbits: its Bloch vector (cos(pi/8), sin(pi/8), 0) is neither an edge nor a face center of the stabilizer octahedron, and the Clifford group acts on Bloch vectors by signed coordinate permutations. Two basis states span it, and with the exact Galois lower bound of cert_sqrtt_galois.py this settles chi(|sqrtT>) = 2. Baseline under the board's rule: 1/2, published, from Theorem 2 of Qassim, Pashayan, and Gosset (arXiv:2106.07740), chi(R_theta^m) = O(2^{m/2}) for every angle theta (explicitly chi(R_theta^(24t+6)) <= 2^(12t+4)). The question the orbit asks is whether a finer phase is dearer per copy than the T state's log_2(3)/4 = 0.3963; the cells that would beat 1/2 are chi(|sqrtT>^5) <= 5 (0.4644) and chi(|sqrtT>^6) <= 7 (0.4680)."
}