magic cat

The m qubits magic cat orbit. Published γ≤log234≈0.3963 through the implied H-type exponent log₂χ(catm)/(m−2).

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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The state

|catm⟩=|T⟩⊗m+|T⟂⟩⊗m2=2−(m−1)/2∑|x| eveni|x|/2|x⟩|T⟩=|0⟩+eiπ/4|1⟩2,|T⟂⟩=Z|T⟩

The magic cat state of Qassim, Pashayan, and Gosset (arXiv:2106.07740, Eq. 3): the equal superposition of |T⟩⊗m and |T⊥⟩⊗m with |T⊥⟩ = Z|T⟩, which in the computational basis is supported on the even-weight strings with phase i|x|/2. This is a family indexed by the qubit count m, not a tensor power: each m is one state, and the schema's m is the number of qubits. |cat₂⟩ = (|00⟩ + i|11⟩)/√2 is a stabilizer state, so the track starts at m = 2.

The family carries the published qubit exponent. Gluing copies of |catm⟩ through the stabilizer bra ⟨cat₂| gives |catl(m−2)+2⟩ in χ(catm)l terms, and χ(T⊗m)/2 ≤ χ(catm) ≤ χ(T⊗m) (their Eq. 4), so a rank-r decomposition of |catm⟩ gives γ ≤ log₂(r)/(m−2) for the H-type orbit; that is the number in the γ column here, and at m = 6, rank 3, it is the published log₂3/4. The values are their Table 1: χ = 1, 2, 2, 3, 3 at m = 2 to 6 (exact) and χ ≤ 6 at m = 7, 8. The decompositions the papers write out (m = 2, 4, 6, the ⟨cat₂| contraction at m = 8, and the ⟨0| projections to m = 3, 5, 7, also in Kissinger, van de Wetering, and Vilmart, arXiv:2202.09202) are replayed here as verified witnesses; the lower bounds at m = 5 to 8 rest on their appendix and are cited. χ(cat₈) ≤ 5 would give γ = 0.3870 and χ(T⊗8) ≤ 10.

Bounds on this orbit

boundmrankγtierattribution
χR(|cat2⟩)≤121—verifiedQassim, Pashayan, and Gosset · arXiv:2106.07740
χR(|cat3⟩)≥232—verified@vprusso · stabrank verify_challenge
χR(|cat3⟩)≤2321.0000verifiedQassim, Pashayan, and Gosset · arXiv:2106.07740
χR(|cat4⟩)≥242—verified@vprusso · stabrank verify_challenge
χR(|cat4⟩)≤2420.5000verifiedQassim, Pashayan, and Gosset · arXiv:2106.07740
χR(|cat5⟩)≥353—verified@MicheldeMessieres · stabrank verify_challenge
χR(|cat5⟩)≤3530.5283verifiedQassim, Pashayan, and Gosset · arXiv:2106.07740
χR(|cat6⟩)≥363—verified@MicheldeMessieres · stabrank verify_challenge
χR(|cat6⟩)≤3630.3962verifiedQassim, Pashayan, and Gosset · arXiv:2106.07740
χR(|cat7⟩)≥373—citedQassim, Pashayan, and Gosset · arXiv:2106.07740
χR(|cat7⟩)≥474—verified@MicheldeMessieres · stabrank verify_challenge
χR(|cat7⟩)≥575—attested · 2/1600 re-run@vprusso · stabrank-research, round 31 (attacks/round31-J-lower-bounds)
χR(|cat7⟩)≤6760.5170verifiedQassim, Pashayan, and Gosset · arXiv:2106.07740
χR(|cat8⟩)≥383—citedQassim, Pashayan, and Gosset · arXiv:2106.07740
χR(|cat8⟩)≥484—verified@MicheldeMessieres · stabrank verify_challenge
χR(|cat8⟩)≥585—attested · 2/1600 re-run@vprusso · stabrank-research, round 31 (attacks/round31-J-lower-bounds)
χR(|cat8⟩)≤6860.4308verifiedQassim, Pashayan, and Gosset · arXiv:2106.07740

What would move it

Beating the published exponent through the implied H-type exponent log₂χ(catm)/(m−2) needs χ ≤ 6 at m=9→ 0.3693.