χR(∣cat⟩⊗3) ≤ 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_3>, checked symbolically

Implied per-copy exponent γ ≤ 1.0000 against a published 0.3962 through the implied H-type exponent log₂χ(catm)/(m−2), no improvement.

Notes

χ(|cat3⟩) ≤ 2 from ⟨0|_4 |cat4⟩ proportional to |cat3⟩ applied to the two-term decomposition of |cat4⟩ (Qassim, Pashayan, and Gosset, appendix, χ(cat_3) ≤ χ(cat_4) ≤ 2; Kissinger, van de Wetering, and Vilmart, Section 4.1, draw the projected decomposition in ZX). Exact with the rank-1 exclusion of bounds/cat-m3-lower-2.json. Witness built by research/constructions/qpg_cat.py by projecting the two cat_4 terms, and verified in exact arithmetic. Implied H-type exponent log_2(2)/1 = 1.

Attribution

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

Submission

JSON
{
  "schema_version": "0.1",
  "orbit": "cat",
  "m": 3,
  "direction": "upper",
  "rank": 2,
  "provenance": {
    "author": "Qassim, Pashayan, and Gosset",
    "reference": "arXiv:2106.07740",
    "method": "literature",
    "date": "2021-06-14",
    "github": []
  },
  "notes": "chi(|cat_3>) <= 2 from <0|_4 |cat_4> proportional to |cat_3> applied to the two-term decomposition of |cat_4> (Qassim, Pashayan, and Gosset, appendix, chi(cat_3) <= chi(cat_4) <= 2; Kissinger, van de Wetering, and Vilmart, Section 4.1, draw the projected decomposition in ZX). Exact with the rank-1 exclusion of bounds/cat-m3-lower-2.json. Witness built by research/constructions/qpg_cat.py by projecting the two cat_4 terms, and verified in exact arithmetic. Implied H-type exponent log_2(2)/1 = 1.",
  "witness": {
    "terms": [
      {
        "k": 2,
        "x0": [
          0,
          0,
          0
        ],
        "W": [
          [
            1,
            0,
            1
          ],
          [
            0,
            1,
            1
          ]
        ],
        "Q": [
          [
            0,
            0
          ],
          [
            0,
            0
          ]
        ],
        "l": [
          0,
          0
        ]
      },
      {
        "k": 0,
        "x0": [
          0,
          0,
          0
        ],
        "W": [],
        "Q": [],
        "l": []
      }
    ],
    "coeffs": [
      "I",
      "1/2 - I/2"
    ]
  }
}