CS — Stabilizer Rank Challenge

The state

| C S ⟩ = C S | + + ⟩ = \frac{| 00 ⟩ + | 01 ⟩ + | 10 ⟩ + i | 11 ⟩}{2}

The controlled-S magic state CS|++⟩, the first two-qubit orbit on the board: m copies live on 2m qubits and γ = log₂(rank)/m is still per copy. CS is the simplest third-level gate on two qubits, CS² = CZ, and Clifford+CS circuits sit between Clifford+T and Clifford+CCZ. The phase i^x₁x₂ is not a Z₄-valued quadratic form with even cross terms, which is exactly why |CS⟩ is not a stabilizer state; CS = ((1+i)/2) I + ((1−i)/2) CZ gives χ(CS) = 2.

χ(CS^⊗2) = 3 exactly, below the product bound 4: |CS⟩^⊗2 = ½|0000⟩ − ((1+i)/2)|+i⟩^⊗4 − ((1−i)/2)(CZ⊗CZ)|−i⟩^⊗4, and rank 2 is excluded exactly over the 36,720 four-qubit stabilizer states in Q(i). So the transversal-number bound 2^τ for hypergraph states is not tight even for two disjoint edges of size two. At three copies the block slice-and-lift excludes rank 3 and the product gives 6, so 4 ≤ χ(CS^⊗3) ≤ 6. No exponent has been published for this orbit; the baseline is the single-copy product bound log₂2 = 1, and the two-copy cell already sits below it at 0.7925. Comparison lines from the literature: the extent exponent log₂(8/5) = 0.678 (F(CS) = 5/8 and ξ(CS) = 8/5 by the Clifford-magic-state theorem of arXiv:1808.00128, an approximate-rank exponent, not a bound on χ), and 3 log₂3/4 = 1.19 for synthesising CS from three T gates.

Bounds on this orbit

bound	m	rank	γ	tier	attribution
$χ_{R} ({\| C S ⟩}^{\otimes 1}) \geq 2$	1	2	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| C S ⟩}^{\otimes 1}) \leq 2$	1	2	1.0000	verified	@vprusso · stabrank docs/notes/new_orbits_design.md
$χ_{R} ({\| C S ⟩}^{\otimes 2}) \geq 3$	2	3	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| C S ⟩}^{\otimes 2}) \leq 3$	2	3	0.7925	verified	@vprusso · stabrank docs/notes/new_orbits_design.md
$χ_{R} ({\| C S ⟩}^{\otimes 3}) \geq 3$	3	3	—	verified	@vprusso · stabrank verify_challenge
$χ_{R} ({\| C S ⟩}^{\otimes 3}) \geq 4$	3	4	—	reproduced	@vprusso · stabrank slice_lift.py
$χ_{R} ({\| C S ⟩}^{\otimes 3}) \leq 6$	3	6	0.8617	verified	@vprusso · stabrank research/constructions/product_witness.py

What would move it

The baseline exponent (single-copy product bound; no published exponent) is already beaten: χ ≤ 3 at m=2 gives γ = 0.7925. Lowering it further needs χ ≤ 8 at m=4→ 0.7500.