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   certificate script asserted: CERTIFIED chi(qubit_sqrtT^1) >= 2, by an argument declared exact throughout (no floating-point margin)

Compute

1 run; 0.003 CPU-hours; 0.003 h wall-clock; on Apple silicon laptop, one core; about 10 s for the certificate; claude-fable-5-1, found the descent argument and wrote the certificate.

Notes

Settles χ(|sqrtT⟩) = 2 with the two-term witness. Every qubit stabilizer state is a scalar times a vector over Z[i], so the span of a decomposition is defined over Q(i) and contains every image of the target under the four automorphisms zeta → zeta^k (k = 1, 5, 9, 13) of Q(zeta_16) fixing i; at one copy the conjugates (1, zeta^k) span C^2, so no single stabilizer state works (directly: the amplitude ratio e^{i π/8} is not a fourth root of unity). Exact throughout: the only computation is the rank of the 2^m x 4 matrix of conjugates in Q(zeta_16), by sympy's algebraic-field arithmetic, and no floating-point margin stands under it. The same certificate prints χ(|sqrtT⟩⊗m) ≥ min(4, m + 1) for m ≤ 6.

Attribution

Vincent Russo · stabrank verify_challenge · Galois descent over Z[i]: the four conjugates of the target under Gal(Q(zeta_16)/Q(i)) span a space of dimension min(4, m + 1); no search

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "qubit_sqrtT",
  "m": 1,
  "direction": "lower",
  "rank": 2,
  "certificate": {
    "script": "verify_challenge/cert_sqrtt_galois.py",
    "expect": "CERTIFIED chi(qubit_sqrtT^1) >= 2",
    "exact": true
  },
  "provenance": {
    "author": "Vincent Russo",
    "reference": "stabrank verify_challenge",
    "method": "Galois descent over Z[i]: the four conjugates of the target under Gal(Q(zeta_16)/Q(i)) span a space of dimension min(4, m + 1); no search",
    "date": "2026-10-21",
    "github": [
      "vprusso"
    ],
    "compute": {
      "cpu_hours": 0.003,
      "wall_clock_hours": 0.003,
      "runs": 1,
      "hardware": "Apple silicon laptop, one core; about 10 s for the certificate",
      "llm": [
        {
          "model": "claude-fable-5-1",
          "role": "found the descent argument and wrote the certificate"
        }
      ]
    }
  },
  "notes": "Settles chi(|sqrtT>) = 2 with the two-term witness. Every qubit stabilizer state is a scalar times a vector over Z[i], so the span of a decomposition is defined over Q(i) and contains every image of the target under the four automorphisms zeta -> zeta^k (k = 1, 5, 9, 13) of Q(zeta_16) fixing i; at one copy the conjugates (1, zeta^k) span C^2, so no single stabilizer state works (directly: the amplitude ratio e^{i pi/8} is not a fourth root of unity). Exact throughout: the only computation is the rank of the 2^m x 4 matrix of conjugates in Q(zeta_16), by sympy's algebraic-field arithmetic, and no floating-point margin stands under it. The same certificate prints chi(|sqrtT>^m) >= min(4, m + 1) for m <= 6."
}