χ_R(∣cat⟩^⊗4) ≤ 2

magic cat orbit, m qubits.

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

verified 2 stabilizer terms reproduce |cat_4>, checked symbolically

Implied per-copy exponent γ ≤ 0.5000 against a published 0.3962 through the implied H-type exponent log₂χ(cat_m)/(m−2), no improvement.

Notes

χ(|cat₄⟩) ≤ 2 from |cat₄⟩ = i|E⟩ + ((1-i)/2) 2^{-1/2}(|0^4> - i|1^4>), |E⟩ the uniform superposition of the even-weight four-bit strings (Qassim, Pashayan, and Gosset, appendix; Kissinger, van de Wetering, and Vilmart, Section 4.1, use it as the 2^{0.25 t} decomposition). Exact with bounds/cat-m4-lower-2.json. Witness built by research/constructions/qpg_cat.py from the two terms as written, and verified in exact arithmetic. Implied H-type exponent log_2(2)/2 = 0.5.

Attribution

Qassim, Pashayan, and Gosset · arXiv:2106.07740 · literature

Submission

JSON

{
  "schema_version": "0.1",
  "orbit": "cat",
  "m": 4,
  "direction": "upper",
  "rank": 2,
  "provenance": {
    "author": "Qassim, Pashayan, and Gosset",
    "reference": "arXiv:2106.07740",
    "method": "literature",
    "date": "2021-06-14",
    "github": []
  },
  "notes": "chi(|cat_4>) <= 2 from |cat_4> = i|E> + ((1-i)/2) 2^{-1/2}(|0^4> - i|1^4>), |E> the uniform superposition of the even-weight four-bit strings (Qassim, Pashayan, and Gosset, appendix; Kissinger, van de Wetering, and Vilmart, Section 4.1, use it as the 2^{0.25 t} decomposition). Exact with bounds/cat-m4-lower-2.json. Witness built by research/constructions/qpg_cat.py from the two terms as written, and verified in exact arithmetic. Implied H-type exponent log_2(2)/2 = 0.5.",
  "witness": {
    "terms": [
      {
        "k": 3,
        "x0": [
          0,
          0,
          0,
          0
        ],
        "W": [
          [
            1,
            0,
            0,
            1
          ],
          [
            0,
            1,
            0,
            1
          ],
          [
            0,
            0,
            1,
            1
          ]
        ],
        "Q": [
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ],
          [
            0,
            0,
            0
          ]
        ],
        "l": [
          0,
          0,
          0
        ]
      },
      {
        "k": 1,
        "x0": [
          0,
          0,
          0,
          0
        ],
        "W": [
          [
            1,
            1,
            1,
            1
          ]
        ],
        "Q": [
          [
            0
          ]
        ],
        "l": [
          3
        ]
      }
    ],
    "coeffs": [
      "I",
      "1/2 - I/2"
    ]
  }
}